MEMORABILIA   MATHEMATICA 


THE  MACMILLAN  COMPANY 

NEW  YORK   •  BOSTON  •   CHICAGO   •  DALLAS 
ATLANTA  •   SAN  FRANCISCO 

MACMILLAN  &  CO.,  LIMITED 

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THE  MACMILLAN  CO.  OF  CANADA,  LTD. 

TORONTO 


Ex  Libris 
C.  K.  OGDEN 


MEMORABILIA  MATHEMATICA 


OR 


THE  PHILOMATH'S  QUOTATION-BOOK 


BY 
ROBERT  EDOUARD  MORITZ,  PH.  D.,  PH.  N.  D. 

PROFESSOR    OF    MATHEMATICS    IN  THE    UNIVERSITY    OF    WASHINGTON 


THE  MACMILLAN  COMPANY 
1914 

All  rights  reserved 


COPYRIGHT,  1914,  BY 
ROBERT  EDOUARD  MORITZ 


(5  A  LIBRARY 

-,  UNIVERSITY  OF  CALIFORNIA 

SANTA  BARBARA 


PREFACE 

EVERY  one  knows  that  the  fine  phrase  "God  geometrizes "  is 
attributed  to  Plato,  but  few  know  where  this  famous  passage  is 
foun  '.  or  the  exact  words  in  which  it  was  first  expressed.  Those 
who,  like  the  author,  have  spent  hours  and  even  days  in  the 
search  of  the  exact  statements,  or  the  exact  references,  of  similar 
famous  passages,  will  not  question  the  timeliness  and  usefulness 
of  a  book  whose  distinct  purpose  it  is  to  bring  together  into  a 
single  volume  exact  quotations,  with  their  exact  references, 
bearing  on  one  of  the  most  time-honored,  and  even  today  the 
most  active  and  most  fruitful  of  all  the  sciences,  the  queen- 
mother  of  all  the  sciences,  that  is,  mathematics. 

It  is  hoped  that  the  present  volume  will  prove  indispensable 
to  every  teacher  of  mathematics,  to  every  writer  on  mathe- 
matics, and  that  the  student  of  mathematics  and  the  related 
sciences  will  find  its  perusal  not  only  a  source  of  pleasure  but  of 
encouragement  and  inspiration  as  well.  The  layman  will  find  it 
a  repository  of  useful  information  covering  a  field  of  knowledge 
which,  owing  to  the  unfamiliar  and  hence  repellant  character  of 
the  language  employed  by  mathematicians,  is  peculiarly  in- 
accessible to  the  general  reader.  No  technical  processes  or 
technical  facility  is  required  to  understand  and  appreciate  the 
wealth  of  ideas  here  set  forth  in  the  words  of  the  world's  great 
thinkers. 

No  labor  has  been  spared  to  make  the  present  volume  worthy 
of  a  place  among  collections  of  a  like  kind  in  other  fields.  Ten 
years  have  been  devoted  to  its  preparation,  years,  which  if  they 
could  have  been  more  profitably,  could  scarcely  have  been  more 
pleasurably  employed.  As  a  result  there  have  been  brought 
together  over  one  thousand  more  or  less  familiar  passages 
pertaining  to  mathematics,  by  poets,  philosophers,  historians, 
statesmen,  scientists,  and  mathematicians.  These  have  been 
gathered  from  over  three  hundred  authors,  and  have  been 

v 


VI  PREFACE 

grouped  under  twenty  heads,  and  cross  indexed  under  nearly 
seven  hundred  topics. 

The  author's  original  plan  was  to  give  foreign  quotations  both 
in  the  original  and  in  translation,  but  with  the  growth  of  mate- 
rial this  plan  was  abandoned  as  infeasible.  It  was  thought  to 
serve  the  best  interest  of  the  greater  number  of  English  readers 
to  give  translations  only,  while  preserving  the  references  to  the 
original  sources,  so  that  the  student  or  critical  reader  may 
readily  consult  the  original  of  any  given  extract.  In  cases  where 
the  translation  is  borrowed  the  translator's  name  is  inserted  in 
brackets  [  ]  immediately  after  the  author's  name.  Brackets  are 
also  used  to  indicate  inserted  words  or  phrases  made  necessary 
to  bring  out  the  context. 

The  absence  of  similar  English  works  has  made  the  author's 
work  largely  that  of  the  pioneer.  Rebie're's  "  Mathematiques  et 
Mathematiciens"  and  Ahrens'  "Scherz  und  Ernst  in  der 
Mathematik"  have  indeed  been  frequently  consulted  but  rather 
with  a  view  to  avoid  overlapping  than  to  receive  aid.  Thus 
certain  topics  as  the  correspondence  of  German  and  French 
mathematicians,  so  excellently  treated  by  Ahrens,  have  pur- 
posely been  omitted.  The  repetitions  are  limited  to  a  small 
number  of  famous  utterances  whose  absence  from  a  work  of  this 
kind  could  scarcely  be  defended  on  any  grounds. 

No  one  can  be  more  keenly  aware  of  the  shortcomings  of  a 
work  than  its  author,  for  none  can  have  so  intimate  an  acquaint- 
ance with  it.  Among  those  of  the  present  work  is  its  incom- 
pleteness, but  it  should  be  borne  in  mind  that  incompleteness  is 
a  necessary  concomitant  of  every  collection  of  whatever  kind. 
Much  less  can  completeness  be  expected  in  a  first  collection, 
made  by  a  single  individual,  in  his  leisure  hours,  and  in  a  field 
which  is  already  boundless  and  is  yet  expanding  day  by  day. 
A  collection  of  great  thoughts,  even  if  complete  today,  would 
be  incomplete  tomorrow.  Again,  if  some  authors  are  quoted 
more  frequently  than  others  of  greater  fame  and  authority,  the 
reason  may  be  sought  not  only  in  the  fact  that  the  writings  of 
some  authors  peculiarly  lent  themselves  to  quotation,  a  quality 
singularly  absent  in  other  writers  of  the  greatest  merit  and 
authority,  but  also  in  this,  that  the  greatest  freedom  has  been 
exercised  in  the  choice  of  selections.  The  author  has  followed 


PREFACE  Vll 

the  bent  of  his  own  fancy  in  collecting  whatever  seemed  to  him 
sufficiently  valuable  because  of  its  content,  its  beauty,  its  origi- 
nality, or  its  terseness,  to  deserve  a  place  in  a  "Memorabilia." 

Great  pains  has  been  taken  to  furnish  exact  readings  and 
references.  In  some  cases  where  a  passage  could  not  be  traced 
to  its  first  source,  the  secondary  source  has  been  given  rather 
than  the  reputed  source.  For  the  same  reason  many  references 
are  to  later  editions  rather  than  to  inaccessible  first  editions. 

The  author  feels  confident  that  this  work  will  be  of  assistance 
to  his  co-workers  in  the  field  of  mathematics  and  allied  fields. 
If  in  addition  it  should  aid  in  a  better  appreciation  of  mathe- 
maticians and  their  work  on  the  part  of  laymen  and  students  in 
other  fields,  the  author's  foremost  aim  in  the  preparation  of  this 
work  will  have  been  achieved. 

ROBERT  EDOUARD  MORITZ, 

September,  1913. 


CONTENTS 

CHAPTER  PAGE 

I.  DEFINITIONS  AND  OBJECT  OF  MATHEMATICS    .      .       1 

II.  THE  NATURE  OF  MATHEMATICS 10 

III.  ESTIMATES  OF  MATHEMATICS 39 

IV.  THE  VALUE  OF  MATHEMATICS 49. 

V.  THE  TEACHING  OF  MATHEMATICS 72 

VI.  STUDY  AND  RESEARCH  IN  MATHEMATICS     ...  86 

VII.  MODERN  MATHEMATICS 108 

VIII.  THE  MATHEMATICIAN 121 

IX.  PERSONS  AND  ANECDOTES  (A-M) 135 

X.  PERSONS  AND  ANECDOTES  (N-Z) 166 

XI.  MATHEMATICS  AS  A  FINE  ART 181 

XII.  MATHEMATICS  AS  A  LANGUAGE 194 

XIII.  MATHEMATICS  AND  LOGIC 201 

XIV.  MATHEMATICS  AND  PHILOSOPHY 209 

XV.  MATHEMATICS  AND  SCIENCE 224 

XVI.  ARITHMETIC 261 

XVII.  ALGEBRA 275 

XVIII.  GEOMETRY 292 

XIX.  THE  CALCULUS  AND  ALLIED  TOPICS  ....  323 

XX.  THE  FUNDAMENTAL  CONCEPTS  OF  TIME  AND  SPACE  345 

XXI.  PARADOXES  AND  CURIOSITIES 364 

INDEX  .  385 


Alles  Gescheite  ist  schon  gedacht  worden;  man  muss  nur  versuchen, 
es  noch  einmaljzu  denken. — -GOETHE. 

Spruche  in  Prosa,  Ethisches,  I.  1. 

A  great  man  quotes  bravely,  and  will  not  draw  on  his  invention  when 
his  memory  serves  him  with  a  word  as  good. — EMERSON. 

Letters  and  Social  Aims,  Quotation  and  Originality. 


MEMORABILIA   MATHEMATICA 


MEMORABILIA   MATHEMATICA 

CHAPTER  I 

DEFINITIONS  AND   OBJECT   OF  MATHEMATICS 

101.  I  think  it  would  be  desirable  that  this  form  of  word 
[mathematics]  should  be  reserved  for  the  applications  of  the 
science,  and  that  we  should  use  mathematic  in  the  singular  to 
denote  the  science  itself,  in  the  same  way  as  we  speak  of  logic, 
rhetoric,  or  (own  sister  to  algebra)  music. — SYLVESTER,  J.  J. 

Presidential  Address  to  the  British  Association, 
Exeter  British  Association  Report  (1869); 
Collected  Mathematical  Papers,  Vol.  2,  p.  659. 

102.  ...  all  the  sciences  which  have  for  their  end  investiga- 
tions concerning  order  and  measure,  are  related  to  mathematics, 
it  being  of  small  importance  whether  this  measure  be  sought  in 
numbers,  forms,  stars,  sounds,  or  any  other  object;  that,  ac- 
cordingly, there  ought  to  exist  a  general  science  which  should 
explain  all  that  can  be  known  about  order  and  measure,  con- 
sidered independently  of  any  application  to  a  particular  subject, 
and  that,  indeed,  this  science  has  its  own  proper  name,  con- 
secrated by  long  usage,  to  wit,  mathematics.    And  a  proof  that 
it  far  surpasses  in  facility  and  importance  the  sciences  which 
depend  upon  it  is  that  it  embraces  at  once  all  the  objects  to 
which  these  are  devoted  and  a  great  many  others  besides;  .  .  . 

DESCARTES. 

Rules  for  the  Direction  of  the  Mind,  Philosophy 
of  D.  [Torrey]  (New  York,  18918),  p.  72. 

103.  [Mathematics]  has  for  its  object  the  indirect  measure- 
ment of  magnitudes,  and  it  purposes  to  determine  magnitudes  by 
each  other,  according  to  the  precise  relations  which  exist  between 
them. — COMTE. 

Positive  Philosophy  [Martineau],  Bk.l,  chap.  1. 


2  MEMORABILIA   MATHEMATICA 

104.  The  business  of  concrete  mathematics  is  to  discover  the 
equations  which  express  the  mathematical  laws  of  the  phenom- 
enon under  consideration;  and  these  equations  are  the  starting- 
point  of  the  calculus;  which  must  obtain  from  them  certain 

quantities  by  means  of  others. — COMTE. 

Positive  Philosophy  [Martineau],  Bk.  1,  chap.  2. 

105.  Mathematics  is  the  science  of  the  connection  of  magni- 
tudes.   Magnitude  is  anything  that  can  be  put  equal  or  unequal 
to  another  thing.    Two  things  are  equal  when  in  every  assertion 
each  may  be  replaced  by  the  other. — GRASSMANN,  HERMANN. 

Stiicke  aus  dem  Lehrbuche  der  Arithmetik, 
Werke  (Leipzig,  1904),  Bd.  2,  p.  298. 

106.  Mathematic  is  either  Pure  or  Mixed:  To  Pure  Mathe- 
matic  belong  those  sciences  which  handle  Quantity  entirely 
severed  from  matter  and  from  axioms  of  natural  philosophy. 
These  are  two,  Geometry  and  Arithmetic;  the  one  handling 
quantity  continued,  the  other  dissevered.  .  .  .  Mixed  Mathe- 
matic has  for  its  subject  some  axioms  and  parts  of  natural 
philosophy,  and  considers  quantity  in  so  far  as  it  assists  to  ex- 
plain, demonstrate  and  actuate  these. — BACON,  FRANCIS. 

De  Augmentis,  Bk.  3;  Advancement 
of  Learning,  Bk.  2. 

107.  The  ideas  which  these  sciences,  Geometry,  Theoretical 
Arithmetic  and  Algebra  involve  extend  to  all  objects  and  changes 
which  we  observe  in  the  external  world;  and  hence  the  considera- 
tion of  mathematical  relations  forms  a  large  portion  of  many  of 
the  sciences  which  treat  of  the  phenomena  and  laws  of  external 
nature,  as  Astronomy,  Optics,  and  Mechanics.    Such  sciences 
are  hence  often  termed  Mixed  Mathematics,  the  relations  of 
space  and  number  being,  in  these  branches  of  knowledge,  com- 
bined with  principles  collected  from  special  observation;  while 
Geometry,  Algebra,  and  the  like  subjects,  which  involve  no 
result  of  experience,  are  called  Pure  Mathematics. 

WHEWELL,  WILLIAM. 
The   Philosophy   of   the   Inductive   Sciences, 
Part  1,  Bk.  2,  chap.  I,  sect.  4.  (London,  1858). 


DEFINITIONS   AND    OBJECTS   OF   MATHEMATICS  3 

108.  Higher  Mathematics  is  the  art  of  reasoning  about 
numerical  relations  between  natural  phenomena;  and  the  sev- 
eral sections  of  Higher  Mathematics  are  different  modes  of 
viewing  these  relations. — MELLOR,  J.  W. 

Higher  Mathematics  for  Students  of  Chemistry 
and  Physics    (New    York,    1902),   Prologue. 


109.  Number,  place,  and  combination  .  .  .  the  three  inter- 
secting but  distinct  spheres  of  thought  to  which  all  mathemati- 
cal ideas  admit  of  being  referred. — SYLVESTER,  J.  J. 

Philosophical  Magazine,  Vol.  24.  (1844), 
p.  285;  Collected  Mathematical  Papers,  Vol.  1, 
p.  91. 


110.  There  are  three  ruling  ideas,  three  so  to  say,  spheres  of 
thought,  which  pervade  the  whole  body  of  mathematical 
science,  to  some  one  or  other  of  which,  or  to  two  or  all  three 
of  them  combined,  every  mathematical  truth  admits  of  be- 
ing referred;  these  are  the  three  cardinal  notions,  of  Number, 
Space  and  Order. 

Arithmetic  has  for  its  object  the  properties  of  number  in  the 
abstract.  In  algebra,  viewed  as  a  science  of  operations,  order 
is  the  predominating  idea.  The  business  of  geometry  is  with  the 
evolution  of  the  properties  of  space,  or  of  bodies  viewed  as 
existing  in  space. — SYLVESTER,  J.  J. 

A  Probationary  Lecture  on  Geometry,  York 
British  Association  Report  (1844),  Part  2; 
Collected  Mathematical  Papers,  Vol.  2,  p.  5. 


111.  The  object  of  pure  mathematics  is  those  relations  which 
may  be  conceptually  established  among  any  conceived  elements 
whatsoever  by  assuming  them  contained  in  some  ordered  mani- 
fold; the  law  of  order  of  this  manifold  must  be  subject  to  our 
choice;  the  latter  is  the  case  in  both  of  the  only  conceivable 
kinds  of  manifolds,  in  the  discrete  as  well  as  in  the  continuous. 

PAPPERITZ,  E. 

Uber  das  System  der  rein  mathematischen 
Wissenschaften,  Jahresbericht  der  Deutschen 
Mathematiker-Vereinigung,  Bd.  1,  p.  86. 


4  MEMORABILIA   MATHEMATICA 

112.  Pure  mathematics  is  not  concerned  with  magnitude. 
It  is  merely  the  doctrine  of  notation  of  relatively  ordered  thought 
operations  which  have  become  mechanical. — NOVALIS. 

Schriften  (Berlin,  1901),  Zweiter  Teil,  p.  282. 

113.  Any   conception   which   is   definitely   and   completely 
determined  by  means  of  a  finite  number  of  specifications,  say 
by  assigning  a  finite  number  of  elements,  is  a  mathematical 
conception.    Mathematics  has  for  its  function  to  develop  the 
consequences  involved  in  the  definition  of  a  group  of  math- 
ematical   conceptions.     Interdependence   and  mutual   logical 
consistency  among  the  members  of  the  group  are  postulated, 
otherwise  the  group  would  either  have  to  be  treated  as  several 
distinct  groups,  or  would  lie  beyond  the  sphere  of  mathematics. 

CHRYSTAL,  GEORGE. 

Encyclopedia  Britannica  (9th  edition),  Article 
11  Mathematics." 

114.  The  purely  formal  sciences,  logic  and  mathematics,  deal 
with  those  relations  which  are,  or  can  be,  independent  of  the 
particular  content  or  the  substance  of  objects.    To  mathematics 
in  particular  fall  those  relations  between  objects  which  involve 
the  concepts  of  magnitude,  of  measure  and  of  number. 

HANKEL,  HERMANN. 

Theorie  der  Complexen  Zahlensysteme,  (Leipzig, 
1867),  p.  1. 

115.  Quantity  is  that  which  is  operated  with  according  to  fixed 
mutually  consistent  laws.     Both  operator  and  operand  must 
derive  their  meaning  from  the  laws  of  operation.    In  the  case  of 
ordinary  algebra  these  are  the  three  laws  already  indicated 
[the  commutative,  associative,  and  distributive  laws],  in  the 
algebra  of  quaternions  the  same  save  the  law  of  commutation 
for  multiplication  and  division,  and  so  on.    It  may  be  questioned 
whether  this  definition  is  sufficient,  and  it  may  be  objected  that 
it  is  vague;  but  the  reader  will  do  well  to  reflect  that  any  defini- 
tion must  include  the  linear  algebras  of  Peirce,  the  algebra  of 
logic,  and  others  that  may  be  easily  imagined,  although  they 
have  not  yet  been  developed.    This  general  definition  of  quan- 


DEFINITIONS   AND   OBJECTS   OF   MATHEMATICS  5 

tity  enables  us  to  see  how  operators  may  be  treated  as  quanti- 
ties, and  thus  to  understand  the  rationale  of  the  so  called  sym- 
bolical methods. — CHRYSTAL,  GEORGE. 

Encyclopedia  Britannica  (9th  edition),  Article 
" Mathematics." 

116.  Mathematics — in  a  strict  sense — is  the  abstract  science 
which  investigates  deductively  the  conclusions  implicit  in  the 
elementary  conceptions  of  spatial  and  numerical  relations. 

MURRAY,  J.  A.  H. 
A  New  English  Dictionary. 

117.  Everything  that  the  greatest  minds  of  all  times  have 
accomplished  toward  the  comprehension  of  forms  by  means  of 
concepts  is  gathered  into  one  great  science,  mathematics. 

HERBART,  J.  F. 

Pestalozzi's  Idee  eines  ABC  der  Anschauung, 
Werke  [Kehrbach],  (Langensalza,  1890),  Bd.  1, 
p.  163. 

118.  Perhaps  the  least  inadequate  description  of  the  general 
scope  of  modern  Pure  Mathematics — I  will  not  call  it  a  defini- 
tion— would  be  to  say  that  it  deals  with  form,  in  a  very  general 
sense  of  the  term;  this  would  include  algebraic  form,  functional 
relationship,  the  relations  of  order  in  any  ordered  set  of  entities 
such  as  numbers,  and  the  analysis  of  the  peculiarities  of  form  of 
groups  of  operations. — HOBSON,  E.  W. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science  (1910);  Nature, 
Vol.  84,  p.  287. 

119.  The  ideal  of  mathematics  should  be  to  erect  a  calculus 
to  facilitate  reasoning  in  connection  with  every  province  of 
thought,  or  of  external  experience,  in  which  the  succession  of 
thoughts,  or  of  events  can  be  definitely  ascertained  and  pre- 
cisely stated.    So  that  all  serious  thought  which  is  not  philos- 
ophy, or  inductive  reasoning,  or  imaginative  literature,  shall  be 
mathematics  developed  by  means  of  a  calculus. 

WHITEHEAD,  A.  N. 
Universal  Algebra  (Cambridge,  1898),  Preface. 


6  MEMORABILIA   MATHEMATICA 

120.  Mathematics  is  the  science  which  draws  necessary  con- 
clusions.— PEIRCE,  BENJAMIN. 

Linear  Associative  Algebra,  American  Journal 
of  Mathematics,  Vol.  4  (1881),  p.  97. 

121.  Mathematics  is  the  universal  art  apodictic. 

SMITH,  W.  B. 

Quoted  by  Keyser,  C.  J.  in  Lectures  on  Science, 
Philosophy  and  Art  (New  York,  1908),  p.  18. 

122.  Mathematics  in  its  widest  signification  is  the  develop- 
ment of  all  types  of  formal,  necessary,  deductive  reasoning. 

WHITEHEAD,  A.  N. 

Universal  Algebra  (Cambridge,  1898),  Preface, 
p.  vi. 

123.  Mathematics  in  general  is  fundamentally  the  science  of 

self-evident  things. — KLEIN,  FELIX. 

Anwendung    der    Differential-und    Integral- 
rechnung  auf  Geometric  (Leipzig,  1902},  p.  26. 

124.  A  mathematical  science  is  any  body  of  propositions 
which  is  capable  of  an  abstract  formulation  and  arrangement 
in  such  a  way  that  every  proposition  of  the  set  after  a  certain 
one  is  a  formal  logical  consequence  of  some  or  all  the  preceding 
propositions.    Mathematics  consists  of  all  such  mathematical 

sciences. — YOUNG,  CHARLES  WESLEY. 

Fundamental  Concepts  of  Algebra  and  Geome- 
try (New  York,  1911),  p.  222. 

125.  Pure  mathematics  is  a  collection  of  hypothetical,  deduc- 
tive theories,  each  consisting  of  a  definite  system  of  primitive, 
undefined,  concepts  or  symbols  and  primitive,  unproved,  but 
self-consistent  assumptions  (commonly  called  axioms)  together 
with  their  logically  deducible  consequences  following  by  rigidly 
deductive  processes  without  appeal  to  intuition. — FITCH,  G.  D. 

The    Fourth    Dimension    simply    Explained 
(New  York,  1910),  p.  58. 

126.  The  whole  of  Mathematics  consists  in  the  organization 
of  a  series  of  aids  to  the  imagination  in  the  process  of  reasoning. 

WHITEHEAD,  A.  N. 
Universal  Algebra  (Cambridge,  1898),  p.  12. 


DEFINITIONS   AND    OBJECTS   OF   MATHEMATICS  7 

127.  Pure  mathematics  consists  entirely  of  such  assevera- 
tions as  that,  if  such  and  such  a  proposition  is  true  of  anything, 
then  such  and  such  another  proposition  is  true  of  that  thing. 
It  is  essential  not  to  discuss  whether  the  first  proposition  is 
really  true,  and  not  to  mention  what  the  anything  is  of  which 
it  is  supposed  to  be  true.  ...     If  our  hypothesis  is  about 
anything  and  not  about  some  one  or  more  particular  things,  then 
our  deductions  constitute  mathematics.     Thus  mathematics 
may  be  defined  as  the  subject  in  which  we  never  know  what 
we  are  talking  about,  nor  whether  what  we  are  saying  is  true. 

RUSSELL,  BERTBAND. 
Recent  Work  on  the  Principles  of  Mathematics, 
International  Monthly,   Vol.  4  (1901),  P-  $4- 

128.  Pure  Mathematics  is  the  class  of  all  propositions  of  the 
form  "p  implies  q,"  where  p  and  q  are  propositions  containing 
one  or  more  variables,  the  same  in  the  two  propositions,  and 
neither  p  nor  q  contains  any  constants  except  logical  constants. 
And  logical  constants  are  all  notions  definable  in  terms  of  the 
following:  Implication,  the  relation  of  a  term  to  a  class  of  which 
it  is  a  member,  the  notion  of  such  that,  the  notion  of  relation, 
and  such  further  notions  as  may  be  involved  in  the  general 
notion  of  propositions  of  the  above  form.    In  addition  to  these, 
Mathematics  uses  a  notion  which  is  not  a  constituent  of  the 
propositions  which  it  considers — namely,  the  notion  of  truth. 

RUSSELL,  BERTRAND. 
Principles  of  Mathematics  (Cambridge,  1903), 
p.L 

129.  The  object  of  pure  Physic  is  the  unfolding  of  the  laws 
of  the  intelligible  world;  the  object  of  pure  Mathematic  that  of 
unfolding  the  laws  of  human  intelligence. — SYLVESTER,  J.  J. 

On  a  theorem  connected  with  Newton's  Rule, 
etc.,  Collected  Mathematical  Papers,  Vol.  3, 
p.  424- 

130.  First  of  all,  we  ought  to  observe,  that  mathematical 
propositions,  properly  so  called,  are  always  judgments  a  priori, 
and  not  empirical,  because  they  carry  along  with  them  necessity, 
which  can  never  be  deduced  from  experience.    If  people  should 


8  MEMORABILIA   MATHEMATICA 

object  to  this,  I  am  quite  willing  to  confine  my  statements  to 
pure  mathematics,  the  very  concept  of  which  implies  that  it 
does  not  contain  empirical,  but  only  pure  knowledge  a  priori. 

KANT,  IMMANUEL. 

Critique  of  Pure  Reason  [Muller],  (New  York, 
1900),  p.  720. 

131.  Mathematics,  the  science  of  the  ideal,   becomes  the 
means  of  investigating,  understanding  and  making  known  the 
world  of  the  real.    The  complex  is  expressed  in  terms  of  the 
simple.    From  one  point  of  view  mathematics  may  be  defined  as 
the  science  of  successive  substitutions  of  simpler  concepts  for 
more  complex.  .  .  . — WHITE,  WILLIAM  F. 

A    Scrap-book   of  Elementary   Mathematics, 
(Chicago,  1908),  p.  215. 

132.  The  critical  mathematician  has  abandoned  the  search 
for  truth.    He  no  longer  flatters  himself  that  his  propositions  are 
or  can  be  known  to  him  or  to  any  other  human  being  to  be 
true;  and  he  contents  himself  with  aiming  at  the  correct,  or  the 
consistent.    The  distinction  is  not  annulled  nor  even  blurred  by 
the  reflection  that  consistency  contains  immanently  a  kind  of 
truth.    He  is  not  absolutely  certain,  but  he  believes  profoundly 
that  it  is  possible  to  find  various  sets  of  a  few  propositions  each 
such  that  the  propositions  of  each  set  are  compatible,  that 
the  propositions  of  each  such  set  imply  other  propositions,  and 
that  the  latter  can  be  deduced  from  the  former  with  certainty. 
That  is  to  say,  he  believes  that  there  are  systems  of  coherent  or 
consistent  propositions,  and  he  regards  it  his  business  to  dis- 
cover such  systems.    Any  such  system  is  a  branch  of  mathe- 
matics.— KEYSER,  C.  J. 

Science,  New  Series,   Vol.  35,  p.  107. 


133.  [Mathematics  is]  the  study  of  ideal  constructions  (often 
applicable  to  real  problems),  and  the  discovery  thereby  of  rela- 
tions between  the  parts  of  these  constructions,  before  unknown. 

PEIRCE,  C.  S. 
Century  Dictionary,  Article  "Mathematics" 


DEFINITIONS   AND   OBJECTS   OF   MATHEMATICS  9 

134.  Mathematics  is  that  form  of  intelligence  in  which  we 
bring  the  objects  of  the  phenomenal  world  under  the  control  of 
the  conception  of  quantity.  [Provisional  definition.] 

HOWISON,  G.  H. 

The  Departments  of  Mathematics,  and  their 
Mutual  Relations;  Journal  of  Speculative 
Philosophy,  Vol.  S,  p.  164. 

136.  Mathematics  is  the  science  of  the  functional  laws  and 
transformations  which  enable  us  to  convert  figured  extension 
and  rated  motion  into  number. — HOWISON,  G.  H. 

The  Departments  of  Mathematics,  and  their 
Mutual  Relations;  Journal  of  Speculative 
Philosophy,  Vol.  5,  p.  170. 


CHAPTER  II 

THE  NATURE   OF  MATHEMATICS 

201.  Mathematics,   from  the  earliest  times  to  which  the 
history  of  human  reason  can  reach,  has  followed,  among  that 
wonderful  people  of  the  Greeks,  the  safe  way  of  science.    But  it 
must  not  be  supposed  that  it  was  as  easy  for  mathematics  as  for 
logic,  in  which  reason  is  concerned  with  itself  alone,  to  find,  or 
rather  to  make  for  itself  that  royal  road.    I  believe,  on  the  con- 
trary, that  there  was  a  long  period  of  tentative  work  (chiefly 
still  among  the  Egyptians),  and  that  the  change  is  to  be  as- 
cribed to  a  revolution,  produced  by  the  happy  thought  of  a  single 
man,  whose  experiments  pointed  unmistakably  to  the  path  that 
had  to  be  followed,  and  opened  and  traced  out  for  the  most 
distant  times  the  safe  way  of  a  science.    The  history  of  that 
intellectual  revolution,  which  was  far  more  important  than  the 
passage  round  the  celebrated  Cape  of  Good  Hope,  and  the 
name  of  its  fortunate  author,   have  not  been  preserved  to 
us.  ...    A  new  light  flashed  on  the  first  man  who  demonstrated 
the  properties  of  the  isosceles  triangle  (whether  his  name  was 
Thales  or  any  other  name),  for  he  found  that  he  had  not  to 
investigate  what  he  saw  hi  the  figure,  or  the  mere  concepts  of 
that  figure,  and  thus  to  learn  its  properties;  but  that  he  had 
to  produce  (by  construction)  what  he  had  himself,  according  to 
concepts  a  priori,  placed  into  that  figure  and  represented  in  it, 
so  that,  in  order  to  know  anything  with  certainty  a  priori,  he 
must  not  attribute  to  that  figure  anything  beyond  what  neces- 
sarily follows  from  what  he  has  himself  placed  into  it,  in  accord- 
ance with  the  concept. — KANT,  IMMANUEL. 

Critique  of  Pure  Reason,  Preface  to  the  Second 
Edition  [Mutter],  (New  York,  1900),  p.  690. 

202.  [When  followed  in  the  proper  spirit],  there  is  no  study  in 
the  world  which  brings  into  more  harmonious  action  all  the 
faculties  of  the  mind  than  the  one  [mathematics]  of  which  I 

10 


THE    NATURE   OF   MATHEMATICS  11 

stand  here  as  the  humble  representative  and  advocate.  There 
is  none  other  which  prepares  so  many  agreeable  surprises  for  its 
followers,  more  wonderful  than  the  transformation  scene  of  a 
pantomime,  or,  like  this,  seems  to  raise  them,  by  successive 
steps  of  initiation  to  higher  and  higher  states  of  conscious 
intellectual  being. — SYLVESTER,  J.  J. 

A  Plea  for  the  Mathematician,  Nature,  Vol.  1, 

p.  261. 

203.  Thought-economy  is  most  highly  developed  in  mathe- 
matics,  that  science  which  has  reached  the  highest  formal 
development,  and  on  which  natural  science  so  frequently  calls 
for  assistance.    Strange  as  it  may  seem,  the  strength  of  mathe- 
matics lies  in  the  avoidance  of  all  unnecessary  thoughts,  in  the 
utmost  economy  of  thought-operations.    The  symbols  of  order, 
which  we  call  numbers,  form  already  a  system  of  wonderful 
simplicity  and  economy.     When  in  the  multiplication  of  a 
number  with  several  digits  we  employ  the  multiplication  table 
and  thus  make  use  of  previously  accomplished  results  rather 
than  to  repeat  them  each  time,  when  by  the  use  of  tables  of 
logarithms  we  avoid  new  numerical  calculations  by  replacing 
them  by  others  long  since  performed,  when  we  employ  deter- 
minants instead  of  carrying  through  from  the  beginning  the 
solution  of  a  system  of  equations,  when  we  decompose  new 
integral  expressions  into  others  that  are  familiar, — we  see  in  all 
this  but  a  faint  reflection  of  the  intellectual  activity  of  a  La- 
grange  or  Cauchy,  who  with  the  keen  discernment  of  a  military 
commander  marshalls  a  whole  troop  of  completed  operations 
in  the  execution  of  a  new  one. — MACH,  E. 

.  Popular-wissenschafliche  Vorlesungen  (1908), 
pp.  224-225. 

204.  Pure  mathematics  proves  itself  a  royal  science  both 
through  its  content  and  form,  which  contains  within  itself  the 
cause  of  its  being  and  its  methods  of  proof.    For  in  complete 
independence  mathematics  creates  for  itself  the  object  of  which 
it  treats,  its  magnitudes  and  laws,  its  formulas  and  symbols. 

DlLLMANN,    E. 

Die  Mathematik  die  Fackeltragerin  einer  neuen 
Zeit  (Stuttgart,  1889),  p.  94. 


12  MEMORABILIA  MATHEMATICA 

206.  The  essence  of  mathematics  lies  in  its  freedom. 

CANTOR,  GEORGE. 
Mathematische  Annalen,  Bd.  21,  p.  564- 

206.  Mathematics  pursues  its  own  course  unrestrained,  not 
indeed  with  an  unbridled  licence  which  submits  to  no  laws,  but 
rather  with  the  freedom  which  is  determined  by  its  own  nature 
and  in  conformity  with  its  own  being. — HANKEL,  HERMANN. 

Die   Entwickelung    der    Mathematik   in   den 
letzten  Jahrhunderten  (Tubingen,  1884),  P-  16. 

207.  Mathematics  is  perfectly  free  in  its  development  and  is 
subject  only  to  the  obvious  consideration,  that  its  concepts 
must  be  free  from  contradictions  in  themselves,  as  well  as 
definitely  and  orderly  related  by  means  of  definitions  to  the 
previously  existing  and  established  concepts. 

CANTOR,  GEORGE. 

Grundlagen  einer  attgemeinen  Manigfaltigkeits- 
lehre  (Leipzig,  1883),  Sect.  8. 

208.  Mathematicians  assume  the  right  to  choose,  within  the 
limits  of  logical  contradiction,  what  path  they  please  in  reaching 

their  results. — ADAMS,  HENRY. 

A  Letter   to  American    Teachers  of  History 
(Washington,  1910),  Introduction,  p.  v. 

209.  Mathematics  is  the  predominant  science  of  our  time;  its 
conquests  grow  daily,  though  without  noise;  he  who  does  not 
employ  it  for  himself,  will  some  day  find  it  employed  against 

himself. — HERBART,  J.  F. 

Werke  [Kehrbach]  (Langensalza,  1890),  Bd.  5, 
p.  105. 

210.  Mathematics  is  not  the  discoverer  of  laws,  for  it  is  not 
induction;  neither  is  it  the  framer  of  theories,  for  it  is  not  hy- 
pothesis; but  it  is  the  judge  over  both,  and  it  is  the  arbiter  to 
which  each  must  refer  its  claims;  and  neither  law  can  rule  nor 
theory  explain  without  the  sanction  of  mathematics. 

PEIRCE,  BENJAMIN. 

Linear  Associative  Algebra,  American  Journal 
of  Mathematics,  Vol.  4  (1881),  p.  97. 


THE   NATURE   OF   MATHEMATICS  13 

211.  Mathematics  is  a  science  continually  expanding;  and  its 
growth,  unlike  some  political  and  industrial  events,  is  attended 
by  universal  acclamation. — WHITE,  H.  S. 

Congress  of  Arts  and  Sciences  (Boston  and 
New  York,  1905),  Vol.  1,  p.  455. 

212.  Mathematics  accomplishes  really  nothing  outside  of  the 
realm  of  magnitude;  marvellous,  however,  is  the  skill  with 
which  it  masters  magnitude  wherever  it  finds  it.    We  recall  at 
once  the  network  of  lines  which  it  has  spun  about  heavens  and 
earth;  the  system  of  lines  to  which  azimuth  and  altitude,  dec- 
lination and  right  ascension,  longitude  and  latitude  are  re- 
ferred; those  abscissas  and  ordinates,  tangents  and  normals, 
circles  of  curvature  and  evolutes;  those  'trigonometric  and 
logarithmic  functions  which  have  been  prepared  in  advance  and 
await  application.     A  look  at  this  apparatus  is  sufficient  to 
show  that  mathematicians  are  not  magicians,  but  that  every- 
thing is  accomplished  by  natural  means;  one  is  rather  impressed 
by  the  multitude  of  skilful  machines,  numerous  witnesses  of  a 
manifold  and  intensely  active  industry,  admirably  fitted  for  the 
acquisition  of  true  and  lasting  treasures. — HERBART,  J.  F. 

Werke  [Kehrbach]    (Langensalza,  1890),  Bd. 
5,  p.  101. 

213.  They   [mathematicians]   only   take   those   things   into 
consideration,  of  which  they  have  clear  and  distinct  ideas, 
designating  them  by  proper,  adequate,  and  invariable  names, 
and  premising  only  a  few  axioms  which  are  most  noted  and 
certain  to  investigate  their  affections  and  draw  conclusions  from 
them,  and  agreeably  laying  down  a  very  few  hypotheses,  such  as 
are  in  the  highest  degree  consonant  with  reason  and  not  to  be 
denied  by  anyone  in  his  right  mind.    In  like  manner  they  assign 
generations  or  causes  easy  to  be  understood  and  readily  ad- 
mitted by  all,  they  preserve   a  most  accurate  order,  every 
proposition  immediately  following  from  what  is  supposed  and 
proved  before,  and  reject  all  things  howsoever  specious  and 
probable  which  can  not  be  inferred  and  deduced  after  the  same 
manner. — BARROW,  ISAAC. 

Mathematical  Lectures  (London,  1734),  P-  66. 


14  MEMORABILIA  MATHEMATICA 

214.  The  dexterous  management  of  terms  and  being  able  to 
fend  and  prove  with  them,  I  know  has  and  does  pass  in  the 
world  for  a  great  part  of  learning;  but  it  is  learning  distinct  from 
knowledge,    for   knowledge    consists   only   in   perceiving   the 
habitudes  and  relations  of  ideas  one  to  another,  which  is  done 
without  words;  the  intervention  of  sounds  helps  nothing  to  it. 
And  hence  we  see  that  there  is  least  use  of  distinction  where 
there  is  most  knowledge:  I  mean  in  mathematics,  where  men 
have  determined  ideas  with  known  names  to  them;  and  so, 
there  being  no  room  for  equivocations,  there  is  no  need  of  dis- 
tinctions.— LOCKE,  JOHN. 

Conduct  of  the  Understanding,  Sect.  81. 

215.  In  mathematics  it  [sophistry]  had  no  place  from  the 
beginning:  Mathematicians  having  had  the  wisdom  to  define 
accurately  the  terms  they  use,  and  to  lay  down,  as  axioms,  the 
first  principles  on  which  their  reasoning  is  grounded.    Accord- 
ingly we  find  no  parties  among  mathematicians,  and  hardly  any 
disputes. — REID,  THOMAS. 

Essays  on  the  Intellectual  Powers  of  Man, 
Essay  1,  chap.  1. 

216.  In  most  sciences  one  generation  tears  down  what  another 
has  built  and  what  one  has  established  another  undoes.     In 
Mathematics  alone  each  generation  builds  a  new  story  to  the 

old  structure. — HANKEL,  HERMANN. 

Die   Entwickelung   der   Mathematik   in   den 
letzten  J ahrhunderten  (Tubingen,  1884),  P-  %5- 

217.  Mathematics,  the  priestess  of  definiteness  and  clear- 
ness.— HERBART,  J.  F. 

Werke  [Kehrbach]  (Langensaha,  1890),  Bd.  1, 
p.  171. 

218.  .  .  .  mathematical  analysis  is  co-extensive  with  nature 
itself,  it  defines  all  perceivable  relations,  measures  times,  spaces, 
forces,  temperatures;  it  is  a  difficult  science  which  forms  but 
slowly,  but  preserves  carefully  every  principle  once  acquired; 
it  increases  and  becomes  stronger  incessantly  amidst  all  the 
changes  and  errors  of  the  human  mind. 


THE   NATURE    OF   MATHEMATICS  15 

Its  chief  attribute  is  clearness;  it  has  no  means  for  expressing 
confused  ideas.  It  compares  the  most  diverse  phenomena  and 
discovers  the  secret  analogies  which  unite  them.  If  matter 
escapes  us,  as  that  of  air  and  light  because  of  its  extreme  tenuity, 
if  bodies  are  placed  far  from  us  in  the  immensity  of  space,  if 
man  wishes  to  know  the  aspect  of  the  heavens  at  successive 
periods  separated  by  many  centuries,  if  gravity  and  heat  act 
in  the  interior  of  the  solid  earth  at  depths  which  will  forever  be 
inaccessible,  mathematical  analysis  is  still  able  to  trace  the 
laws  of  these  phenomena.  It  renders  them  present  and  measur- 
able, and  appears  to  be  the  faculty  of  the  human  mind  destined 
to  supplement  the  brevity  of  life  and  the  imperfection  of  the 
senses,  and  what  is  even  more  remarkable,  it  follows  the  same 
course  in  the  study  of  all  phenomena;  it  explains  them  in  the 
same  language,  as  if  in  witness  to  the  unity  and  simplicity  of  the 
plan  of  the  universe,  and  to  make  more  manifest  the  unchange- 
able order  which  presides  over  all  natural  causes. — FOURIER,  J. 

Theorie  Analytique  de  la  Chaleur,  Discours 
Preliminaire. 

219.  Let  us  now  declare  the  means  whereby  our  understand- 
ing can  rise  to  knowledge  without  fear  of  error.  There  are  two 
such  means:  intuition  and  deduction.  By  intuition  I  mean  not 
the  varying  testimony  of  the  senses,  nor  the  deductive  judgment 
of  imagination  naturally  extravagant,  but  the  conception  of  an 
attentive  mind  so  distinct  and  so  clear  that  no  doubt  remains 
to  it  with  regard  to  that  which  it  comprehends;  or,  what  amounts 
to  the  same  thing,  the  self-evidencing  conception  of  a  sound  and 
attentive  mind,  a  conception  which  springs  from  the  light  of 
reason  alone,  and  is  more  certain,  because  more  simple,  than 
deduction  itself.  .  .  . 

It  may  perhaps  be  asked  why  to  intuition  we  add  this  other 
mode  of  knowing,  by  deduction,  that  is  to  say,  the  process 
which,  from  something  of  which  we  have  certain  knowledge, 
draws  consequences  which  necessarily  follow  therefrom.  But 
we  are  obliged  to  admit  this  second  step;  for  there  are  a  great 
many  things  which,  without  being  evident  of  themselves,  never- 
theless bear  the  marks  of  certainty  if  only  they  are  deduced  from 
true  and  incontestable  principles  by  a  continuous  and  uninter- 


16  MEMORABILIA   MATHEMATICA 

rupted  movement  of  thought,  with  distinct  intuition  of  each 
thing;  just  as  we  know  that  the  last  link  of  a  long  chain  holds  to 
the  first,  although  we  can  not  take  in  with  one  glance  of  the  eye 
the  intermediate  links,  provided  that,  after  having  run  over 
them  in  succession,  we  can  recall  them  all,  each  as  being  joined  to 
its  fellows,  from  the  first  up  to  the  last.  Thus  we  distinguish 
intuition  from  deduction,  inasmuch  as  in  the  latter  case  there  is 
conceived  a  certain  progress  or  succession,  while  it  is  not  so  in 
the  former;  .  .  .  whence  it  follows  that  primary  propositions, 
derived  immediately  from  principles,  may  be  said  to  be  known, 
according  to  the  way  we  view  them,  now  by  intuition,  now  by 
deduction;  although  the  principles  themselves  can  be  known 
only  by  intuition,  the  remote  consequences  only  by  deduction. 

DESCARTES. 

Rules  for  the  Direction  of  the  Mind,  Philosophy 
of  D.  [Torrey]  (New  York,  1892),  pp.  64,  65. 

220.  Analysis  and  natural  philosophy  owe  their  most  impor- 
tant discoveries  to  this  fruitful  means,  which  is  called  induction. 
Newton  was  indebted  to  it  for  his  theorem  of  the  binomial  and 
the  principle  of  universal  gravity. — LAPLACE. 

A  Philosophical  Essay  on  Probabilities  [Tru- 
scott  and  Emory]  (New  York  1902),  p.  176. 

221.  There  is  in  every  step  of  an  arithmetical  or  algebraical 
calculation  a  real  induction,  a  real  inference  from  facts  to  facts, 
and  what  disguises  the  induction  is  simply  its  comprehensive 
nature,  and  the  consequent  extreme  generality  of  its  language. 

MILL,  J.  S. 
System  of  Logic,  Bk.  2,  chap.  6,  2. 

222.  It  would  appear  that  Deductive  and  Demonstrative 
Sciences  are  all,  without  exception,  Inductive  Sciences:  that 
their  evidence  is  that  of  experience,  but  that  they  are  also,  in 
virtue  of  the  peculiar  character  of  one  indispensable  portion  of 
the  general  formulae  according  to  which  their  inductions  are 
made,  Hypothetical  Sciences.    Their  conclusions  are  true  only 
upon  certain  suppositions,  which  are,  or  ought  to  be,  approxi- 
mations to  the  truth,  but  are  seldom,  if  ever,  exactly  true;  and 


THE   NATURE   OF  MATHEMATICS  17 

to  this  hypothetical  character  is  to  be  ascribed  the  peculiar 
certainty,  which  is  supposed  to  be  inherent  in  demonstration. 

MILL,  J.  S. 
System  of  Logic,  Bk.  2,  chap.  6,  1. 

223.  The  peculiar  character  of  mathematical  truth  is,  that 
it  is  necessarily  and  inevitably  true;  and  one  of  the  most  im- 
portant lessons  which  we  learn  from  our  mathematical  studies 
is  a  knowledge  that  there  are  such  truths,  and  a  familiarity 
with  their  form  and  character. 

This  lesson  is  not  only  lost,  but  read  backward,  if  the  student 
is  taught  that  there  is  no  such  difference,  and  that  mathematical 
truths  themselves  are  learned  by  experience. — WHEWELL,  W. 
Thoughts  on  the  Study  of  Mathematics.    Prin- 
ciples of  English  University  Education  (London, 
1838). 

224.  These  sciences,  Geometry,  Theoretical  Arithmetic  and 
Algebra,  have  no  principles  besides  definitions  and  axioms,  and 
no  process  of  proof  but  deduction;  this  process,  however,  assum- 
ing a  most  remarkable  character;  and  exhibiting  a  combination 
of  simplicity  and  complexity,  of  rigour  and  generality,  quite 
unparalleled  in  other  subjects. — WHEWELL,  W. 

The   Philosophy   of  the   Inductive   Sciences, 
Part  1,  Bk.  2,  chap.  1,  sect.  2  (London,  1858). 

225.  The  apodictic  quality  of  mathematical  thought,  the 
certainty  and  correctness  of  its  conclusions,  are  due,  not  to  a 
special  mode  of  ratiocination,  but  to  the  character  of  the  con- 
cepts with  which  it  deals.     What  is  that  distinctive  charac- 
teristic?     I    answer:    precision,    sharpness,    completeness*    of 
definition.    But  how  comes  your  mathematician  by  such  com- 
pleteness?   There  is  no  mysterious  trick  involved;  some  ideas 
admit  of  such  precision,  others  do  not;  and  the  mathematician  is 

one  who  deals  with  those  that  do. — KEYSER,  C.  J. 

The  Universe  and  Beyond;  Hibbert  Journal, 
Vol.  3  (1904-1905),  p.  809. 

226.  The  reasoning  of  mathematicians  is  founded  on  certain 
and  infallible  principles.    Every  word  they  use  conveys  a  deter- 

*  i.  e.,  in  terms  of  the  absolutely  clear  and  indefinable. 


18  MEMORABILIA   MATHEMATICA 

minate  idea,  and  by  accurate  definitions  they  excite  the  same 
ideas  in  the  mind  of  the  reader  that  were  in  the  mind  of  the 
writer.  When  they  have  defined  the  terms  they  intend  to  make 
use  of,  they  premise  a  few  axioms,  or  self-evident  principles, 
that  every  one  must  assent  to  as  soon  as  proposed.  They  then 
take  for  granted  certain  postulates,  that  no  one  can  deny  them, 
such  as,  that  a  right  line  may  be  drawn  from  any  given  point  to 
another,  and  from  these  plain,  simple  principles  they  have 
raised  most  astonishing  speculations,  and  proved  the  extent  of 
the  human  mind  to  be  more  spacious  and  capacious  than  any 
other  science. — ADAMS,  JOHN. 

Diary,  Works  (Boston,  1850),  Vol.  2,  p.  21. 

227.  It  may  be  observed  of  mathematicians  that  they  only 
meddle  with  such  things  as  are  certain,  passing  by  those  that 
are  doubtful  and  unknown.     They  profess  not  to  know  all 
things,  neither  do  they  affect  to  speak  of  all  things.    What  they 
know  to  be  true,  and  can  make  good  by  invincible  arguments, 
that  they  publish  and  insert  among  their  theorems.    Of  other 
things  they  are  silent  and  pass  no  judgment  at  all,  choosing  rather 
to  acknowledge  their  ignorance,  than  affirm  anything  rashly. 
They  affirm  nothing  among  their  arguments  or  assertions  which 
is  not  most  manifestly  known  and  examined  with  utmost 
rigour,  rejecting  all  probable  conjectures  and  little  witticisms. 
They  submit  nothing  to  authority,  indulge  no  affection,  detest 
subterfuges  of  words,  and  declare  their  sentiments,  as  in  a  court 
of  justice,  without  passion,  without  apology;  knowing  that  their 
reasons,  as  Seneca  testifies  of  them,  are  not  brought  to  persuade, 
but  to  compel. — BARROW,  ISAAC. 

Mathematical  Lectures  (London,  1734),  p.  64- 

228.  What  is  exact  about  mathematics  but  exactness?    And 
is  not  this  a  consequence  of  the  inner  sense  of  truth? — GOETHE. 

Spruche  in  Prosa,  Natur,  6,  948. 

229.  .  .  .  the  three  positive  characteristics  that  distinguish 
mathematical  knowledge  from  other  knowledge  .  .  .  may  be 
briefly  expressed  as  follows:  first,  mathematical  knowledge  bears 
more  distinctly  the  imprint  of  truth  on  all  its  results  than  any 
other  kind  of  knowledge;  secondly,  it  is  always  a  sure  prelimi- 


THE   NATURE    OF   MATHEMATICS  19 

nary  step  to  the  attainment  of  other  correct  knowledge;  thirdly, 
it  has  no  need  of  other  knowledge. — SCHUBERT,  H. 

Mathematical  Essays  and  Recreations  (Chicago, 
1898),  p.  35, 

230.  It  is  now  necessary  to  indicate  more  definitely  the  reason 
why  mathematics  not  only  carries  conviction  in  itself,  but  also 
transmits  conviction  to  the  objects  to  which  it  is  applied.  The 
reason  is  found,  first  of  all,  in  the  perfect  precision  with  which  the 
elementary  mathematical  concepts  are  determined;  in  this 
respect  each  science  must  look  to  its  own  salvation  ....  But 
this  is  not  all.  As  soon  as  human  thought  attempts  long  chains 
of  conclusions,  or  difficult  matters  generally,  there  arises  not 
only  the  danger  of  error  but  also  the  suspicion  of  error,  because 
since  all  details  cannot  be  surveyed  with  clearness  at  the  same 
instant  one  must  in  the  end  be  satisfied  with  a  belief  that  nothing 
has  been  overlooked  from  the  beginning.  Every  one  knows  how 
much  this  is  the  case  even  in  arithmetic,  the  most  elmenetary 
use  of  mathematics.  No  one  would  imagine  that  the  higher 
parts  of  mathematics  fare  better  in  this  respect;  on  the  con- 
trary, in  more  complicated  conclusions  the  uncertainty  and 
suspicion  of  hidden  errors  increases  in  rapid  progression.  How 
does  mathematics  manage  to  rid  itself  of  this  inconvenience 
which  attaches  to  it  in  the  highest  degree?  By  making  proofs 
more  rigorous?  By  giving  new  rules  according  to  which  the 
old  rules  shall  be  applied?  Not  in  the  least.  A  very  great  un- 
certainty continues  to  attach  to  the  result  of  each  single  com- 
putation. But  there  are  checks.  In  the  realm  of  mathematics 
each  point  may  be  reached  by  a  hundred  different  ways;  and  if 
each  of  a  hundred  ways  leads  to  the  same  point,  one  may  be  sure 
that  the  right  point  has  been  reached.  A  calculation  without  a 
check  is  as  good  as  none.  Just  so  it  is  with  every  isolated  proof 
in  any  speculative  science  whatever;  the  proof  may  be  ever  so 
ingenious,  and  ever  so  perfectly  true  and  correct,  it  will  still  fail 
to  convince  permanently.  He  will  therefore  be  much  deceived, 
who,  in  metaphysics,  or  in  psychology  which  depends  on  meta- 
physics, hopes  to  see  his  greatest  care  in  the  precise  determina- 
tion of  the  concepts  and  in  the  logical  conclusions  rewarded  by 
conviction,  much  less  by  success  in  transmitting  conviction  to 


20  MEMORABILIA   MATHEMATICA 

others.  Not  only  must  the  conclusions  support  each  other, 
without  coercion  or  suspicion  of  subreption,  but  in  all  matters 
originating  in  experience,  or  judging  concerning  experience,  the 
results  of  speculation  must  be  verified  by  experience,  not  only 
superficially,  but  in  countless  special  cases. — HERBAHT,  J.  F. 

Werke  [Kehrbach]   (Langensalza,  1890),  Bd. 

5,  p.  105. 

231.  [In    mathematics]    we    behold    the    conscious    logical 
activity  of  the  human  mind  in  its  purest  and  most  perfect  form. 
Here  we  learn  to  realize  the  laborious  nature  of  the  process,  the 
great  care  with  which  it  must  proceed,  the  accuracy  which  is 
necessary  to  determine  the  exact  extent  of  the  general  proposi- 
tions arrived  at,  the  difficulty  of  forming  and  comprehending 
abstract  concepts;  but  here  we  learn  also  to  place  confidence  in 
the  certainty,  scope  and  fruitfulness  of  such  intellectual  activity. 

HELMHOLTZ,  H. 

Ueber  das  Verhaltniss  der  Naturwissenschaften 
zur  Gesammtheit  der  Wissenschaft,  Vortrdge 
und  Reden,  Bd.  1  (1896),  p.  176. 

232.  It  is  true  that  mathematics,  owing  to  the  fact  that  its 
whole  content  is  built  up  by  means  of  purely  logical  deduction 
from  a  small  number  of  universally  comprehended  principles, 
has  not  unfittingly  been  designated  as  the  science  of  the  self- 
evident  [Selbstverstandlichen].    Experience  however,  shows  that 
for  the  majority  of  the  cultured,  even  of  scientists,  mathematics 
remains  the  science  of  the  incomprehensible  [Unversta'ndlichen]. 

PRINGSHEIM,  ALFRED. 
Ueber    Wert    und    angeblichen    Unwert    der 
Mathematik,     Jahresbericht    der     Deutschen 
Mathematiker  Vereinigung  (1904),  P-  357. 

233.  Mathematical  reasoning  is  deductive  in  the  sense  that 
it  is  based  upon  definitions  which,  as  far  as  the  validity  of  the 
reasoning  is  concerned  (apart  from  any  existential  import), 
needs  only  the  test  of  self-consistency.     Thus  no  external 
verification  of  definitions  is  required  in  mathematics,  as  long  as 

it  is  considered  merely  as  mathematics. — WHITEHEAD,  A.  N. 
Universal  Algebra   (Cambridge,  1898),  Pref- 
ace, p.  vi. 


THE   NATURE   OF   MATHEMATICS  21 

234.  The  mathematician  pays  not  the  least  regard  either  to 
testimony  or  conjecture,  but  deduces  everything  by  demon- 
strative reasoning,  from  his  definitions  and  axioms.     Indeed, 
whatever  is  built  upon  conjecture,  is  improperly  called  science; 
for  conjecture  may  beget  opinion,  but  cannot  produce  knowl- 
edge.— REID,  THOMAS. 

Essays  on  the  Intellectual  Powers  of  Man, 
Essay  1,  chap.  8. 

235.  ...  for  the  saving  the  long  progression  of  the  thoughts 
to  remote  and  first  principles  in  every  case,  the  mind  should 
provide  itself  several  stages;  that  is  to  say,  intermediate  prin- 
ciples, which  it  might  have  recourse  to  in  the  examining  those 
positions  that  come  in  its  way.    These,  though  they  are  not 
self-evident  principles,  yet,  if  they  have  been  made  out  from 
them  by  a  wary  and  unquestionable  deduction,  may  be  de- 
pended on  as  certain  and  infallible  truths,  and  serve  as  unques- 
tionable truths  to  prove  other  points  depending  upon  them,  by 
a  nearer  and  shorter  view  than  remote  and  general  maxims.  .  .  . 
And  thus  mathematicians  do,  who  do  not  in  every  new  problem 
run  it  back  to  the  first  axioms  through  all  the  whole  train  of 
intermediate  propositions.     Certain  theorems  that  they  have 
settled  to  themselves  upon  sure  demonstration,  serve  to  resolve 
to  them  multitudes  of  propositions  which  depend  on  them,  and 
are  as  firmly  made  out  from  thence  as  if  the  mind  went  afresh 
over  every  link  of  the  whole  chain  that  tie  them  to  first  self- 
evident  principles. — LOCKE,  JOHN. 

The  Conduct  of  the  Understanding,  Sect.  21 . 

236.  Those  intervening  ideas,  which  serve  to  show  the  agree- 
ment of  any  two  others,  are  called  proofs;  and  where  the  agree- 
ment or  disagreement  is  by  this  means  plainly  and  clearly 
perceived,  it  is  called  demonstration;  it  being  shown  to  the 
understanding,  and  the  mind  made  to  see  that  it  is  so.    A  quick- 
ness in  the  mind  to  find  out  these  intermediate  ideas,  (that  shall 
discover  the  agreement  or  disagreement  of  any  other)  and  to 
apply  them  right,  is,  I  suppose,  that  which  is  called  sagacity. 

LOCKE,  JOHN. 

An  Essay  concerning  Human  Understanding, 
Bk.  6,  chaps.  2,  3. 


22  MEMORABILIA   MATHEMATICA 

237.  .  .  .  the  speculative  propositions  of  mathematics  do  not 
relate  to  facts;  ...  all  that  we  are  convinced  of  by  any  demon- 
stration in  the  science,  is  of  a  necessary  connection  subsisting 
between  certain  suppositions  and  certain  conclusions.     When 
we  find  these  suppositions  actually  take  place  in  a  particular 
instance,  the  demonstration  forces  us  to  apply  the  conclusion. 
Thus,  if  I  could  form  a  triangle,  the  three  sides  of  which  were 
accurately  mathematical  lines,  I  might  affirm  of  this  individual 
figure,  that  its  three  angles  are  equal  to  two  right  angles;  but,  as 
the  imperfection  of  my  senses  puts  it  out  of  my  power  to  be,  in 
any  case,  certain  of  the  exact  correspondence  of  the  diagram 
which  I  delineate,  with  the  definitions  given  in  the  elements  of 
geometry,  I  never  can  apply  with  confidence  to  a  particular 
figure,  a  mathematical  theorem.    On  the  other  hand,  it  appears 
from  the  daily  testimony  of  our  senses  that  the  speculative 
truths  of  geometry  may  be  applied  to  material  objects  with  a 
degree  of  accuracy  sufficient  for  the  purposes  of  life;  and  from 
such  applications  of  them,  advantages  of  the  most  important 
kind  have  been  gained  to  society. — STEWART,  DUGALD. 

Elements  of  the  Philosophy  of  the  Human 
Mind,  Part  8,  chap.  1,  sect.  8. 

238.  No  process  of  sound  reasoning  can  establish  a  result  not 
contained  in  the  premises. — MELLOR,  J.  W. 

Higher  Mathematics  for  Students  of  Chemistry 
and  Physics  (New  York,  1902),  p.  2. 

239.  ...  we  cannot  get  more  out  of  the  mathematical  mill 
than  we  put  into  it,  though  we  may  get  it  in  a  form  infinitely 
more  useful  for  our  purpose. — HOPKINSON,  JOHN. 

James  Forrest  Lecture,  1894- 

240.  The  iron  labor  of  conscious  logical  reasoning  demands 
great  perseverance  and  great  caution;  it  moves  on  but  slowly, 
and  is  rarely  illuminated  by  brilliant  flashes  of  genius.    It  knows 
little  of  that  facility  with  which  the  most  varied  instances  come 
thronging  into  the  memory  of  the  philologist  or  historian. 
Rather  is  it  an  essential  condition  of  the  methodical  progress  of 
mathematical  reasoning  that  the  mind  should  remain  concen- 


THE   NATURE    OF   MATHEMATICS  23 

trated  on  a  single  point,  undisturbed  alike  by  collateral  ideas  on 
the  one  hand,  and  by  wishes  and  hopes  on  the  other,  and  moving 
on  steadily  in  the  direction  it  has  deliberately  chosen. 

HELMHOLTZ,  H. 

Ueber  das  Verhdltniss  der  Naturwissenschaften 
zur  Gesammtheit  der  Wissenschaft,  Vortrdge 
und  Reden,  Bd.  1  (1896),  p.  178. 

241.  If  it  were  always  necessary  to  reduce  everything  to 
intuitive  knowledge,  demonstration  would  often  be  insufferably 
prolix.    This  is  why  mathematicians  have  had  the  cleverness  to 
divide  the  difficulties  and  to  demonstrate  separately  the  inter- 
vening propositions.    And  there  is  art  also  in  this;  for  as  the 
mediate  truths  (which  are  called  lemmas,  since  they  appear 
to  be  a  digression)  may  be  assigned  in  many  ways,  it  is  well,  in 
order  to  aid  the  understanding  and  memory,  to  choose  of  them 
those  which  greatly  shorten  the  process,  and  appear  memorable 
and  worthy  in  themselves  of  being  demonstrated.     But  there  is 
another  obstacle,  viz. :  that  it  is  not  easy  to  demonstrate  all  the 
axioms,   and   to   reduce   demonstrations   wholly   to   intuitive 
knowledge.    And  if  we  had  chosen  to  wait  for  that,  perhaps  we 
should  not  yet  have  the  science  of  geometry. — LEIBNITZ,  G.  W. 

New  Essay  on  Human  Understanding  [Lang- 
ley],  Bk.  4,  chaps.  2,  8. 

242.  In  Pure  Mathematics,  where  all  the  various  truths  are 
necessarily  connected  with  each  other,  (being  all  necessarily 
connected  with  those  hypotheses  which  are  the  principles  of  the 
science),   an  arrangement  is  beautiful  in  proportion  as  the 
principles  are  few;  and  what  we  admire  perhaps  chiefly  in  the 
science,  is  the  astonishing  variety  of  consequences  which  may 
be  demonstrably  deduced  from  so  small  a  number  of  premises. 

STEWART,  DUGALD. 

The  Elements  of  the  Philosophy  of  the  Human 
Mind,  Part  3,  chap.  1,  sect.  3. 

243.  Whenever  ...  a  controversy  arises  in  mathematics, 
the  issue  is  not  whether  a  thing  is  true  or  not,  but  whether  the 
proof  might  not  be  conducted  more  simply  in  some  other  way,  or 
whether  the  proposition  demonstrated  is  sufficiently  important 


24  MEMORABILIA  MATHEMATICA 

for  the  advancement  of  the  science  as  to  deserve  especial 
enunciation  and  emphasis,  or  finally,  whether  the  proposition  is 
not  a  special  case  of  some  other  and  more  general  truth  which  is 

as  easily  discovered. — SCHUBERT,  H. 

Mathematical  Essays  and  Recreations  (Chicago, 
1898),  p.  28. 

244.  .  .  .  just  as  the  astronomer,  the  physicist,  the  geologist, 
or  other  student  of  objective  science  looks  about  in  the  world 
of  sense,  so,  not  metaphorically  speaking  but  literally,  the  mind 
of  the  mathematician  goes  forth  in  the  universe  of  logic  in  quest 
of  the  things  that  are  there;  exploring  the  heights  and  depths 
for  facts — ideas,  classes,  relationships,  implications,  and  the 
rest;   observing  the  minute  and  elusive  with   the   powerful 
microscope  of  his  Infinitesimal  Analysis;  observing  the  elusive 
and  vast  with  the  limitless  telescope  of  his  Calculus  of  the 
Infinite;    making   guesses   regarding  the   order  and   internal 
harmony  of   the   data  observed  and  collocated;  testing  the 
hypotheses,  not  merely  by  the  complete  induction  peculiar  to 
mathematics,  but,  like  his  colleagues  of  the  outer  world,  resort- 
ing also  to  experimental  tests  and  incomplete  induction;  fre- 
quently finding  it  necessary,  in  view  of  unforeseen  disclosures,  to 
abandon  one  hopeful  hypothesis  or  to  transform  it  by  retrench- 
ment or  by  enlargement: — thus,  in  his  own  domain,  matching, 
point  for  point,  the  processes,  methods  and  experience  familiar 
to  the  devotee  of  natural  science. — KEYSER,  CASSIUS,  J. 

Lectures  on  Science,  Philosophy  and  Art  (New 
York,  1908),  p.  26. 

245.  That  mathematics   "do  not   cultivate  the  power  of 
generalization,"  .  .  .  will  be  admitted  by  no  person  of  compe- 
tent knowledge,  except  in  a  very  qualified  sense.    The  generaliza- 
tions of  mathematics,  are,  no  doubt,  a  different  thing  from  the 
generalizations  of  physical  science;  but  in  the  difficulty  of  seizing 
them,  and  the  mental  tension  they  require,  they  are  no  con- 
temptible preparation  for  the  most  arduous  efforts  of  the 
scientific  mind.    Even  the  fundamental  notions  of  the  higher 
mathematics,  from  those  of  the  differential  calculus  upwards  are 
products  of  a  very  high  abstraction.  ...     To  perceive  the 
mathematical  laws  common  to  the  results  of  many  mathematical 


THE   NATURE    OF   MATHEMATICS  25 

operations,  even  in  so  simple  a  case  as  that  of  the  binomial 
theorem,  involves  a  vigorous  exercise  of  the  same  faculty  which 
gave  us  Kepler's  laws,  and  rose  through  those  laws  to  the  theory 
of  universal  gravitation.  Every  process  of  what  has  been  called 
Universal  Geometry — the  great  creation  of  Descartes  and  his 
successors,  in  which  a  single  train  of  reasoning  solves  whole 
classes  of  problems  at  once,  and  others  common  to  large  groups 
of  them — is  a  practical  lesson  in  the  management  of  wide 
generalizations,  and  abstraction  of  the  points  of  agreement  from 
those  of  difference  among  objects  of  great  and  confusing  diver- 
sity, to  which  the  purely  inductive  sciences  cannot  furnish 
many  superior.  Even  so  elementary  an  operation  as  that  of 
abstracting  from  the  particular  configuration  of  the  triangles  or 
other  figures,  and  the  relative  situation  of  the  particular  lines  or 
points,  in  the  diagram  which  aids  the  apprehension  of  a  common 
geometrical  demonstration,  is  a  very  useful,  and  far  from  being 
always  an  easy,  exercise  of  the  faculty  of  generalization  so 
strangely  imagined  to  have  no  place  or  part  in  the  processes  of 
mathematics. — MILL,  JOHN  STUART. 

An  Examination  of  Sir  William  Hamilton's 
Philosophy  (London,  1878),  pp.  612,  613. 

246.  When  the  greatest  of  American  logicians,  speaking  of 
the  powers  that  constitute  the  born  geometrician,  had  named 
Conception,    Imagination,    and    Generalization,    he    paused. 
Thereupon  from  one  of  the  audience  there  came  the  challenge, 
"What  of  reason?"    The  instant  response,  not  less  just  than 
brilliant,  was:  "Ratiocination — that  is  but  the  smooth  pave- 
ment on  which  the  chariot  rolls." — KEYSER,  C.  J. 

Lectures  on  Science,  Philosophy  and  Art  (New 
York,  1908),  p.  31. 

247.  .  .  .  the  reasoning  process  [employed  in  mathematics] 
is  not  different  from  that  of  any  other  branch  of  knowledge,  .  .  . 
but  there  is  required,  and  in  a  great  degree,  that  attention  of 
mind  which  is  in  some  part  necessary  for  the  acquisition  of  all 
knowledge,  and  in  this  branch  is  indispensably  necessary.    This 
must  be  given  in  its  fullest  intensity;  .  .  .  the  other  elements 
especially  characteristic  of  a  mathematical  mind  are  quickness 


26  MEMORABILIA   MATHEMATICA 

in  perceiving  logical  sequence,  love  of  order,  methodical  arrange- 
ment and  harmony,   distinctness  of  conception. — PRICE,   B. 

Treatise  on  Infinitesimal  Calculus  (Oxford, 
1868),  Vol.  3,  p.  6. 

248.  Histories  make  men  wise;  poets,  witty;  the  mathematics, 
subtile;  natural  philosophy,  deep;  moral,  grave;  logic  and  rheto- 
ric, able  to  contend. — BACON,  FRANCIS. 

Essays,  Of  Studies. 

249.  The  Mathematician  deals  with  two  properties  of  objects 
only,  number  and  extension,  and  all  the  inductions  he  wants 
have  been  formed  and  finished  ages  ago.    He  is  now  occupied 
with  nothing  but  deduction  and  verification. — HUXLEY,  T.  H. 

On  the  Educational  Value  of  the  Natural  His- 
tory Sciences;  Lay  Sermons,  Addresses  and 
Reviews;  (New  York,  1872),  p.  87. 

250.  [Mathematics]  is  that  [subject]  which  knows  nothing  of 
observation,  nothing  of  experiment,  nothing  of  induction,  noth- 
ing of  causation. — HUXLEY,  T.  H. 

The  Scientific  Aspects  of  Positivism,  Fortnightly 
Review  (1898);  Lay  Sermons,  Addresses  and 
Reviews,  (New  York,  1872),  p.  169. 

251.  We  are  told  that  "Mathematics  is  that  study  which 
knows  nothing  of  observation,  nothing  of  experiment,  nothing 
of  induction,  nothing  of  causation."    I  think  no  statement  could 
have  been  made  more  opposite  to  the  facts  of  the  case;  that 
mathematical  analysis  is  constantly  invoking  the  aid  of  new 
principles,  new  ideas,  and  new  methods,  not  capable  of  being 
defined  by  any  form  of  words,  but  springing  direct  from  the 
inherent  powers  and  activities  of  the  human  mind,  and  from 
continually  renewed  introspection  of  that  inner  world  of  thought 
of  which  the  phenomena  are  as  varied  and  require  as  close  atten- 
tion to  discern  as  those  of  the  outer  physical  world  (to  which 
the  inner  one  in  each  individual  man  may,  I  think,  be  conceived 
to  stand  somewhat  in  the  same  relation  of  correspondence  as  a 
shadow  to  the  object  from  which  it  is  projected,  or  as  the  hollow 
palm  of  one  hand  to  the  closed  fist  which  it  grasps  of  the  other), 
that  it  is  unceasingly  calling  forth  the  faculties  of  observation 


THE   NATURE    OF   MATHEMATICS  27 

and  comparison,  that  one  of  its  principal  weapons  is  induction, 
that  it  has  frequent  recourse  to  experimental  trial  and  verifica- 
tion, and  that  it  affords  a  boundless  scope  for  the  exercise  of  the 
highest  efforts  of  the  imagination  and  invention. 

SYLVESTER,  J.  J. 

Presidential  Address  to  British  Association, 
Exeter  British  Association  Report  (1869), 
pp.  1-9.;  Collected  Mathematical  Papers, 
Vol.  2,  p.  654. 

252.  The  actual  evolution  of  mathematical  theories  proceeds 
by  a  process  of  induction  strictly  analogous  to  the  method  of 
induction  employed  in  building  up  the  physical  sciences;  obser- 
vation, comparison,  classification,  trial,  and  generalisation  are 
essential  in  both  cases.  Not  only  are  special  results,  obtained 
independently  of  one  another,  frequently  seen  to  be  really  in- 
cluded in  some  generalisation,  but  branches  of  the  subject 
which  have  been  developed  quite  independently  of  one  another 
are  sometimes  found  to  have  connections  which  enable  them  to 
be  synthesised  in  one  single  body  of  doctrine.  The  essential 
nature  of  mathematical  thought  manifests  itself  in  the  dis- 
cernment of  fundamental  identity  in  the  mathematical  aspects 
of  what  are  superficially  very  different  domains.  A  striking 
example  of  this  species  of  immanent  identity  of  mathematical 
form  was  exhibited  by  the  discovery  of  that  distinguished 
mathematician  .  .  .  Major  MacMahon,  that  all  possible  Latin 
squares  are  capable  of  enumeration  by  the  consideration  of 
certain  differential  operators.  Here  we  have  a  case  in  which  an 
enumeration,  which  appears  to  be  not  amenable  to  direct  treat- 
ment, can  actually  be  carried  out  in  a  simple  manner  when  the 
underlying  identity  of  the  operation  is  recognised  with  that 
involved  in  certain  operations  due  to  differential  operators,  the 
calculus  of  which  belongs  superficially  to  a  wholly  different 
region  of  thought  from  that  relating  to  Latin  squares. 

HOBSON,  E.  W. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science  (1910);  Nature, 
Vol.  84,  p.  290. 

253.  It  has  been  asserted  .  .  .  that  the  power  of  observation 
is  not  developed  by  mathematical  studies;  while  the  truth  is, 


28  MEMORABILIA   MATHEMATICA 

that;  from  the  most  elementary  mathematical  notion  that 

arises  in  the  mind  of  a  child  to  the  farthest  verge  to  which 

mathematical  investigation  has  been  pushed  and  applied,  this 

power  is  in  constant  exercise.    By  observation,  as  here  used,  can 

only  be  meant  the  fixing  of  the  attention  upon  objects  (physical 

or  mental)  so  as  to  note  distinctive  peculiarities — to  recognize 

resemblances,  differences,  and  other  relations.     Now  the  first 

mental  act  of  the  child  recognizing  the  distinction  between  one 

and  more  than  one,  between  one  and  two,  two  and  three,  etc.,  is 

exactly  this.     So,  again,  the  first  geometrical  notions  are  as 

pure  an  exercise  of  this  power  as  can  be  given.    To  know  a 

straight  line,  to  distinguish  it  from  a  curve;  to  recognize  a 

triangle  and  distinguish  the  several  forms — what  are  these,  and 

all  perception  of  form,  but  a  series  of  observations?    Nor  is  it 

alone  in  securing  these  fundamental  conceptions  of  number  and 

form  that  observation  plays  so  important  a  part.     The  very 

genius  of  the  common  geometry  as  a  method  of  reasoning — a 

system  of  investigation — is,  that  it  is  but  a  series  of  observations. 

The  figure  being  before  the  eye  in  actual  representation,  or 

before  the  mind  in  conception,  is  so  closely  scrutinized,  that  all 

its  distinctive  features  are  perceived;  auxiliary  lines  are  drawn 

(the  imagination  leading  in  this),  and  a  new  series  of  inspections 

is  made;  and  thus,  by  means  of  direct,  simple  observations,  the 

investigation  proceeds.    So  characteristic  of  common  geometry 

is  this  method  of  investigation,  that  Comte,  perhaps  the  ablest 

of  all  writers  upon  the  philosophy  of  mathematics,  is  disposed 

to  class  geometry,  as  to  its  method,  with  the  natural  sciences, 

being  based  upon  observation.     Moreover,  when  we  consider 

applied  mathematics,  we  need  only  to  notice  that  the  exercise 

of  this  faculty  is  so  essential,  that  the  basis  of  all  such  reasoning, 

the  very  material  with  which  we  build,  have  received  the  name 

observations.     Thus  we  might  proceed  to  consider  the  whole 

range  of  the  human  faculties,  and  find  for  the  most  of  them 

ample  scope  for  exercise  in  mathematical  studies.    Certainly, 

the  memory  will  not  be  found  to  be  neglected.    The  very  first 

steps  in  number — counting,  the  multiplication  table,  etc.,  make 

heavy  demands  on  this  power;  while  the  higher  branches  require 

the  memorizing  of  formulas  which  are  simply  appalling  to  the 

uninitiated.     So  the  imagination,  the  creative  faculty  of  the 


THE   NATURE   OF   MATHEMATICS  29 

mind,  has  constant  exercise  in  all  original  mathematical  in- 
vestigations, from  the  solution  of  the  simplest  problems  to  the 
discovery  of  the  most  recondite  principle;  for  it  is  not  by  sure, 
consecutive  steps,  as  many  suppose,  that  we  advance  from  the 
known  to  the  unknown.  The  imagination,  not  the  logical 
faculty,  leads  in  this  advance.  In  fact,  practical  observation  is 
often  in  advance  of  logical  exposition.  Thus,  in  the  discovery  of 
truth,  the  imagination  habitually  presents  hypotheses,  and 
observation  supplies  facts,  which  it  may  require  ages  for  the 
tardy  reason  to  connect  logically  with  the  known.  Of  this 
truth,  mathematics,  as  well  as  all  other  sciences,  affords  abun- 
dant illustrations.  So  remarkably  true  is  this,  that  today  it  is 
seriously  questioned  by  the  majority  of  thinkers,  whether  the 
sublimest  branch  of  mathematics, — the  infinitesimal  calculus — 
has  anything  more  than  an  empirical  foundation,  mathemati- 
cians themselves  not  being  agreed  as  to  its  logical  basis.  That 
the  imagination,  and  not  the  logical  faculty,  leads  in  all  original 
investigation,  no  one  who  has  ever  succeeded  in  producing  an 
original  demonstration  of  one  of  the  simpler  propositions  of 
geometry,  can  have  any  doubt.  Nor  are  induction,  analogy,  the 
scrutinization  of  premises  or  the  search  for  them,  or  the  balancing 
of  probabilities,  spheres  of  mental  operations  foreign  to  mathe- 
matics. No  one,  indeed,  can  claim  pre-eminence  for  mathe- 
matical studies  in  all  these  departments  of  intellectual  culture, 
but  it  may,  perhaps,  be  claimed  that  scarcely  any  department  of 
science  affords  discipline  to  so  great  a  number  of  faculties,  and 
that  none  presents  so  complete  a  gradation  in  the  exercise  of 
these  faculties,  from  the  first  principles  of  the  science  to  the 
farthest  extent  of  its  applications,  as  mathematics. 

OLNEY,  EDWARD. 

Kiddle  and  Schem's  Encyclopedia  of  Education, 
(New   York,   1877),  Article   "Mathematics." 

254.  The  opinion  appears  to  be  gaining  ground  that  this  very 
general  conception  of  functionality,  born  on  mathematical 
ground,  is  destined  to  supersede  the  narrower  notion  of  causa- 
tion, traditional  in  connection  with  the  natural  sciences.  As 
an  abstract  formulation  of  the  idea  of  determination  in  its  most 
general  sense,  the  notion  of  functionality  includes  and  tran- 


30  MEMORABILIA   MATHEMATICA 

scends  the  more  special  notion  of  causation  as  a  one-sided 
determination  of  future  phenomena  by  means  of  present  condi- 
tions; it  can  be  used  to  express  the  fact  of  the  subsumption  under 
a  general  law  of  past,  present,  and  future  alike,  in  a  sequence  of 
phenomena.  From  this  point  of  view  the  remark  of  Huxley 
that  Mathematics  "knows  nothing  of  causation"  could  only  be 
taken  to  express  the  whole  truth,  if  by  the  term  "causation"  is 
understood  "efficient  causation."  The  latter  notion  has,  how- 
ever, in  recent  times  been  to  an  increasing  extent  regarded  as 
just  as  irrelevant  in  the  natural  sciences  as  it  is  in  Mathematics; 
the  idea  of  thorough-going  determinancy,  in  accordance  with 
formal  law,  being  thought  to  be  alone  significant  in  either 
domain. — HOBSON,  E.  W. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science  (1910);  Nature, 
Vol.  84,  p.  290. 

255.  Most,  if  not  all,  of  the  great  ideas  of  modern  mathe- 
matics have  had  their  origin  in  observation.  Take,  for  instance, 
the  arithmetical  theory  of  forms,  of  which  the  foundation  was 
laid  in  the  diophantine  theorems  of  Fermat,  left  without  proof 
by  their  author,  which  resisted  all  efforts  of  the  myriad-minded 
Euler  to  reduce  to  demonstration,  and  only  yielded  up  their 
cause  of  being  when  turned  over  in  the  blow-pipe  flame  of 
Gauss's  transcendent  genius;  or  the  doctrine  of  double  periodi- 
city, which  resulted  from  the  observation  of  Jacobi  of  a  purely 
analytical  fact  of  transformation;  or  Legendre's  law  of  reci- 
procity; or  Sturm's  theorem  about  the  roots  of  equations,  which, 
as  he  informed  me  with  his  own  lips,  stared  him  in  the  face  in 
the  midst  of  some  mechanical  investigations  connected  (if  my 
memory  serves  me  right)  with  the  motion  of  compound  pendu- 
lums; or  Huyghen's  method  of  continued  fractions,  charac- 
terized by  Lagrange  as  one  of  the  principal  discoveries  of  that 
great  mathematician,  and  to  which  he  appears  to  have  been 
led  by  the  construction  of  his  Planetary  Automaton;  or  the 
new  algebra,  speaking  of  which  one  of  my  predecessors  (Mr. 
Spottiswoode)  has  said,  not  without  just  reason  and  authority, 
from  this  chair,  "that  it  reaches  out  and  indissolubly  connects 
itself  each  year  with  fresh  branches  of  mathematics,  that  the 
theory  of  equations  has  become  almost  new  through  it,  alge- 


THE   NATURE   OF   MATHEMATICS  31 

braic  geometry  transfigured  in  its  light,  that  the  calculus  of 
variations,  molecular  physics,  and  mechanics"  (he  might,  if 
speaking  at  the  present  moment,  go  on  to  add  the  theory  of 
elasticity  and  the  development  of  the  integral  calculus)  "have 

all  felt  its  influence." — SYLVESTER,  J.  J. 

A  Plea  for  the  Mathematician,  Nature,  Vol.  1 , 
p.  238;  Collected  Mathematical  Papers,  Vol.  2, 
pp.  655,  656. 

256.  The  ability  to  imagine  relations  is  one  of  the  most 
indispensable  conditions  of  all  precise  thinking.     No  subject 
can  be  named,  in  the  investigation  of  which  it  is  not  impera- 
tively needed;  but  it  can  be  nowhere  else  so  thoroughly  acquired 
as  in  the  study  of  mathematics. — FISKE,  JOHN. 

Darwinism  and  other  Essays  (Boston,  1893), 
p.  296. 

257.  The  great  science  [mathematics]  occupies  itself  at  least 
just  as  much  with  the  power  of  imagination  as  with  the  power 

of  logical  conclusion. — HERBART,  F.  J. 

Pestalozzi's  Idee  eines  ABC  der  Anschauung. 
Werke  [Kehrbach]  (Langensaltza,  1890),  Bd.l, 
p.  174. 

258.  The  moving  power  of  mathematical  invention  is  not 
reasoning  but  imagination. — DE  MORGAN,  A. 

Quoted  in  Graves'  Life  of  Sir  W.  R.  Hamilton, 
Vol.  3  (1889),  p.  219. 

259.  There  is  an  astonishing  imagination,  even  in  the  science 
of  mathematics.  .  .  .    We  repeat,  there  was  far  more  imagina- 
tion in  the  head  of  Archimedes  than  in  that  of  Homer. 

VOLTAIRE. 

A  Philosophical  Dictionary  (Boston,  1881), 
Vol.  3,  p.  40'  Article  "Imagination." 

260.  As  the  prerogative  of  Natural  Science  is  to  cultivate  a 
taste  for  observation,  so  that  of  Mathematics  is,  almost  from 
the  starting  point,  to  stimulate  the  faculty  of  invention. 

SYLVESTER,  J.  J. 

A  Plea  for  the  Mathematician,  Nature,  Vol.  1 , 
p.  261;  Collected  Mathematical  Papers,  Vol.  2 
(Cambridge,  1908),  p.  717. 


32  MEMORABILIA   MATHEMATICA 

261.  A   marvellous    newtrality    have   these   things   mathe- 
maticall,   and   also   a   strange   participation   between   things 
supernaturall,  immortall,  intellectuall,  simple  and  indivisible, 
and  things  naturall,  mortall,  sensible,  componded  and  divisible. 

DEE,  JOHN. 
Euclid  (1570),  Preface. 

262.  Mathematics  stands  forth  as  that  which  unites,  mediates 
between  Man  and  Nature,  inner  and  outer  world,  thought  and 
perception,  as  no  other  subject  does. — FROEBEL. 

[Herford  translation]  (London,  1893),  Vol.  1, 
p.  84. 

263.  The  intrinsic  character  of  mathematical  research  and 
knowledge  is  based  essentially  on  three  properties:  first,  on  its 
conservative  attitude  towards  the  old  truths  and  discoveries  of 
mathematics;  secondly,  on  its  progressive  mode  of  development, 
due  to  the  incessant  acquisition  of  new  knowledge  on  the  basis 
of  the  old;  and  thirdly,  on  its  self-sufficiency  and  its  consequent 
absolute  independence. — SCHUBERT,  H. 

Mathematical  Essays  and  Recreations  (Chicago, 
1898),  p.  27. 

264.  Our  science,  hi  contrast  with  others,  is  not  founded  on  a 
single  period  of  human  history,  but  has  accompanied  the  devel- 
opment of  culture  through  all  its  stages.     Mathematics  is  as 
much  interwoven  with  Greek  culture  as  with  the  most  modern 
problems  in  Engineering.     She  not  only  lends  a  hand  to  the 
progressive  natural  sciences  but  participates  at  the  same  time 
in  the  abstract  investigations  of  logicians  and  philosophers. 

KLEIN,  F. 

Klein  und  Riecke:  Ueber  angewandte  Mathe- 
matik  und  Physik  (1900),  p.  228. 

265.  There  is  probably  no  other  science  which  presents  such 
different  appearances  to  one  who  cultivates  it  and  to  one  who 
does  not,  as  mathematics.    To  this  person  it  is  ancient,  venera- 
ble, and  complete;  a  body  of  dry,  irrefutable,  unambiguous 
reasoning.     To  the  mathematician,   on  the  other  hand,  his 
science  is  yet  in  the  purple  bloom  of  vigorous  youth,  everywhere 


THE   NATURE    OF   MATHEMATICS  33 

stretching  out  after  the  "attainable  but  unattained"  and  full 
of  the  excitement  of  nascent  thoughts;  its  logic  is  beset  with 
ambiguities,  and  its  analytic  processes,  like  Bunyan's  road,  have 
a  quagmire  on  one  side  and  a  deep  ditch  on  the  other  and  branch 
off  into  innumerable  by-paths  that  end  in  a  wilderness. 

CHAPMAN,  C.  H. 

Bulletin  American  Mathematical  Society,  Vol. 
2  (First  series),  p.  61. 

266.  Mathematical  science  is  in  my  opinion  an  indivisible 
whole,  an  organism  whose  vitality  is  conditioned  upon  the 
connection  of  its  parts.     For  with  all  the  variety  of  mathe- 
matical knowledge,  we  are  still  clearly  conscious  of  the  simi- 
larity of  the  logical  devices,  the  relationship  of  the  ideas  in 
mathematics  as  a  whole  and  the  numerous  analogies  in  its 
different  departments.     We  also  notice  that,   the  farther  a 
mathematical  theory  is  developed,  the  more  harmoniously  and 
uniformly  does  its  construction  proceed,  and  unsuspected  rela- 
tions are  disclosed  between  hitherto  separated  branches  of  the 
science.    So  it  happens  that,  with  the  extension  of  mathematics, 
its  organic  character  is  not  lost  but  manifests  itself  the  more 
clearly. — HILBERT,  D. 

Mathematical   Problems,    Bulletin   American 
Mathematical  Society,  Vol.  8,  p.  418. 

267.  The  mathematics  have   always  been  the  implacable 
enemies  of  scientific  romances. — ARAGO. 

Oeuvres  (1855),  t.  8,  p.  498. 

-  268.  Those  skilled  in  mathematical  analysis  know  that  its 
object  is  not  simply  to  calculate  numbers,  but  that  it  is  also 
employed  to  find  the  relations  between  magnitudes  which  cannot 
be  expressed  in  numbers  and  between  functions  whose  law  is  not 
capable  of  algebraic  expression. — COURNOT,  AUGUSTIN. 

Mathematical    Theory   of  the   Principles   of 
Wealth  [Bacon,  N.  T.],  (New  York,  1897),  p.  3. 

269.  Coterminous  with  space  and  coeval  with  time  is  the 
Kingdom  of  Mathematics;  within  this  range  her  dominion  is 
supreme;  otherwise  than  according  to  her  order  nothing  can 
exist;  in  contradiction  to  her  laws  nothing  takes  place.  On  her 


34  MEMORABILIA  MATHEMATICA 

mysterious  scroll  is  to  be  found  written  for  those  who  can  read  it 
that  which  has  been,  that  which  is,  and  that  which  is  to  come. 
Everything  material  which  is  the  subject  of  knowledge  has 
number,  order,  or  position;  and  these  are  her  first  outlines  for  a 
sketch  of  the  universe.  If  our  feeble  hands  cannot  follow  out  the 
details,  still  her  part  has  been  drawn  with  an  unerring  pen,  and 
her  work  cannot  be  gainsaid.  So  wide  is  the  range  of  mathe- 
matical sciences,  so  indefinitely  may  it  extend  beyond  our 
actual  powers  of  manipulation  that  at  some  moments  we  are 
inclined  to  fall  down  with  even  more  than  reverence  before  her 
majestic  presence.  But  so  strictly  limited  are  her  promises  and 
powers,  about  so  much  that  we  might  wish  to  know  does  she 
offer  no  information  whatever,  that  at  other  moments  we  are 
fain  to  call  her  results  but  a  vain  thing,  and  to  reject  them  as  a 
stone  where  we  had  asked  for  bread.  If  one  aspect  of  the  sub- 
ject encourages  our  hopes,  so  does  the  other  tend  to  chasten  our 
desires,  and  he  is  perhaps  the  wisest,  and  in  the  long  run  the 
happiest,  among  his  fellows,  who  has  learned  not  only  this 
science,  but  also  the  larger  lesson  which  it  directly  teaches, 
namely,  to  temper  our  aspirations  to  that  which  is  possible,  to 
moderate  our  desires  to  that  which  is  attainable,  to  restrict  our 
hopes  to  that  of  which  accomplishment,  if  not  immediately 
practicable,  is  at  least  distinctly  within  the  range  of  conception. 

SPOTTISWOODE,  W. 

Quoted    in    Sonnenschein's    Encyclopedia    of 
Education  (London,  1906),  p.  208. 

270.  But  it  is  precisely  mathematics,  and  the  pure  science 
generally,  from  which  the  general  educated  public  and  independ- 
ent students  have  been  debarred,  and  into  which  they  have  only 
rarely  attained  more  than  a  very  meagre  insight.  The  reason 
of  this  is  twofold.  In  the  first  place,  the  ascendant  and  consecu- 
tive character  of  mathematical  knowledge  renders  its  results 
absolutely  insusceptible  of  presentation  to  persons  who  are 
unacquainted  with  what  has  gone  before,  and  so  necessitates 
on  the  part  of  its  devotees  a  thorough  and  patient  exploration  of 
the  field  from  the  very  beginning,  as  distinguished  from  those 
sciences  which  may,  so  to  speak,  be  begun  at  the  end,  and  which 
are  consequently  cultivated  with  the  greatest  zeal.  The  second 


THE   NATURE   OF   MATHEMATICS  35 

reason  is  that,  partly  through  the  exigencies  of  academic  instruc- 
tion, but  mainly  through  the  martinet  traditions  of  antiquity 
and  the  influence  of  mediaeval  logic-mongers,  the  great  bulk  of 
the  elementary  text-books  of  mathematics  have  unconsciously 
assumed  a  very  repellant  form, — something  similar  to  what  is 
termed  in  the  theory  of  protective  mimicry  in  biology  "the 
terrifying  form."  And  it  is  mainly  to  this  formidableness  and 
touch-me-not  character  of  exterior,  concealing  withal  a  harmless 
body,  that  the  undue  neglect  of  typical  mathematical  studies  is 
to  be  attributed. — McCoRMACK,  T.  J. 

Preface  to  De  Morgan's  Elementary  Illustra- 
tions of  the  Differential  and  Integral  Calculus 
(Chicago,  1899). 

271.  Mathematics  in  gross,  it  is  plain,  are  a  grievance  in 
natural  philosophy,  and  with  reason:  for  mathematical  proofs, 
like  diamonds,  are  hard  as  well  as  clear,  and  will  be  touched  with 
nothing  but  strict  reasoning.    Mathematical  proofs  are  out  of 
the  reach  of  topical  arguments;  and  are  not  to  be  attacked  by 
the  equivocal  use  of  words  or  declaration,  that  make  so  great  a 
part  of  other  discourses, — nay,  even  of  controversies. 

LOCKE,  JOHN. 
Second  Reply  to  the  Bishop  of  Worcester. 

272.  The  belief  that  mathematics,  because  it  is  abstract,  be- 
cause it  is  static  and  cold  and  gray,  is  detached  from  life,  is  a 
mistaken  belief.     Mathematics,  even  in  its  purest  and  most 
abstract  estate,  is  not  detached  from  life.    It  is  just  the  ideal 
handling  of  the  problems  of  life,  as  sculpture  may  idealize  a 
human  figure  or  as  poetry  or  painting  may  idealize  a  figure  or 
a  scene.     Mathematics  is  precisely  the  ideal  handling  of  the 
problems  of  life,  and  the  central  ideas  of  the  science,  the  great 
concepts  about  which  its  stately  doctrines  have  been  built  up, 
are  precisely  the  chief  ideas  with  which  life  must  always  deal  and 
which,  as  it  tumbles  and  rolls  about  them  through  time  and 
space,  give  it  its  interests  and  problems,  and  its  order  and 
rationality.    That  such  is  the  case  a  few  indications  will  suffice 
to  show.    The  mathematical  concepts  of  constant  and  variable 
are  represented  familiarly  in  life  by  the  notions  of  fixedness  and 
change.     The  concept  of  equation  or  that  of  an  equational 


36  MEMORABILIA   MATHEMATICA 

system,  imposing  restriction  upon  variability,  is  matched  in 
life  by  the  concept  of  natural  and  spiritual  law,  giving  order  to 
what  were  else  chaotic  change  and  providing  partial  freedom  in 
lieu  of  none  at  all.  What  is  known  in  mathematics  under  the 
name  of  limit  is  everywhere  present  in  life  in  the  guise  of  some 
ideal,  some  excellence  high-dwelling  among  the  rocks,  an  "ever 
flying  perfect"  as  Emerson  calls  it,  unto  which  we  may  approxi- 
mate nearer  and  nearer,  but  which  we  can  never  quite  attain, 
save  in  aspiration.  The  supreme  concept  of  functionality  finds 
its  correlate  in  life  in  the  all-pervasive  sense  of  interdependence 
and  mutual  determination  among  the  elements  of  the  world. 
What  is  known  in  mathematics  as  transformation — that  is, 
lawful  transfer  of  attention,  serving  to  match  in  orderly  fashion 
the  things  of  one  system  with  those  of  another — is  conceived 
in  life  as  a  process  of  transmutation  by  which,  in  the  flux  of  the 
world,  .the  content  of  the  present  has  come  out  of  the  past  and 
in  its  turn,  in  ceasing  to  be,  gives  birth  to  its  successor,  as  the 
boy  is  father  to  the  man  and  as  things,  in  general,  become  what 
they  are  not.  The  mathematical  concept  of  invariance  and  that 
of  infinitude,  especially  the  imposing  doctrines  that  explain 
their  meanings  and  bear  their  names — What  are  they  but 
mathematicizations  of  that  which  has  ever  been  the  chief  of 
life's  hopes  and  dreams,  of  that  which  has  ever  been  the  object 
of  its  deepest  passion  and  of  its  dominant  enterprise,  I  mean  the 
finding  of  the  worth  that  abides,  the  finding  of  permanence  in 
the  midst  of  change,  and  the  discovery  of  a  presence,  in  what  has 
seemed  to  be  a  finite  world,  of  being  that  is  infinite?  It  is  need- 
less further  to  multiply  examples  of  a  correlation  that  is  so 
abounding  and  complete  as  indeed  to  suggest  a  doubt  whether  it 
be  juster  to  view  mathematics  as  the  abstract  idealization  of 
life  than  to  regard  life  as  the  concrete  realization  of  mathematics. 

KEYSER,  C.  J. 

The  Humanization  of  the  Teaching  of  Mathe- 
matics; Science,  Neiv  Series,  Vol.  85,  pp.  643- 
646. 

273.  Mathematics,  like  dialectics,  is  an  organ  of  the  inner 
higher  sense;  hi  its  execution  it  is  an  art  like  eloquence.  Both 
alike  care  nothing  for  the  content,  to  both  nothing  is  of  value 
but  the  form.  It  is  immaterial  to  mathematics  whether  it 


THE   NATURE    OF   MATHEMATICS  37 

computes  pennies  or  guineas,  to  rhetoric  whether  it  defends 
truth  or  error. — GOETHE. 

Wilhelm  Meisters  Wanderjahre,  Zweites  Buck. 

274.  The  genuine  spirit  of  Mathesis  is  devout.    No  intellec- 
tual pursuit  more  truly  leads  to  profound  impressions  of  the 
existence  and  attributes  of  a  Creator,  and  to  a  deep  sense  of 
our  filial  relations  to  him,  than  the  study  of  these  abstract 
sciences.     Who  can  understand  so  well  how  feeble  are  our 
conceptions  of  Almighty  Power,  as  he  who  has  calculated  the 
attraction  of  the  sun  and  the  planets,  and  weighed  in  his  balance 
the  irresistible  force  of  the  lightning?    Who  can  so  well  under- 
stand how  confused  is  our  estimate  of  the  Eternal  Wisdom,  as 
he  who  has  traced  out  the  secret  laws  which  guide  the  hosts  of 
heaven,  and  combine  the  atoms  on  earth?    Who  can  so  well 
understand  that  man  is  made  in  the  image  of  his  Creator,  as  he 
who  has  sought  to  frame  new  laws  and  conditions  to  govern 
imaginary  worlds,  and  found  his  own  thoughts  similar  to  those 
on  which  his  Creator  has  acted? — HILL,  THOMAS. 

The    Imagination    in    Mathematics;    North 
American  Review,  Vol.  85,  p.  226. 

275.  .  .  .  what  is  physical  is  subject  to  the  laws  of  mathe- 
matics, and  what  is  spiritual  to  the  laws  of  God,  and  the  laws  of 
mathematics  are  but  the  expression  of  the  thoughts  of  God. 

HILL,  THOMAS. 

The    Uses   of  Mathesis;   Bibliotheca  Sacra, 
Vol.  82,  p.  523. 

276.  It  is  in  the  inner  world  of  pure  thought,  where  all  entia 
dwell,  where  is  every  type  of  order  and  manner  of  correlation  and 
variety  of  relationship,  it  is  in  this  infinite  ensemble  of  eternal 
verities  whence,  if  there  be  one  cosmos  or  many  of  them,  each 
derives  its  character  and  mode  of  being, — it  is  there  that  the 
spirit  of  Diathesis  has  its  home  and  its  life. 

Is  it  a  restricted  home,  a  narrow  life,  static  and  cold  and  grey 
with  logic,  without  artistic  interest,  devoid  of  emotion  and  mood 
and  sentiment?  That  world,  it  is  true,  is  not  a  world  of  solar 
light,  not  clad  in  the  colours  that  liven  and  glorify  the  things 
of  sense,  but  it  is  an  illuminated  world,  and  over  it  all  and  every- 


38  MEMORABILIA   MATHEMATICA 

where  throughout  are  hues  and  tints  transcending  sense,  painted 
there  by  radiant  pencils  of  psychic  light,  the  light  in  which  it 
lies.  It  is  a  silent  world,  and,  nevertheless,  in  respect  to  the 
highest  principle  of  art — the  interpenetration  of  content  and 
form,  the  perfect  fusion  of  mode  and  meaning — it  even  sur- 
passes music.  In  a  sense,  it  is  a  static  world,  but  so,  too,  are 
the  worlds  of  the  sculptor  and  the  architect.  The  figures,  how- 
ever, which  reason  constructs  and  the  mathematic  vision  be- 
holds, transcend  the  temple  and  the  statue,  alike  in  simplicity 
and  in  intricacy,  in  delicacy  and  in  grace,  in  symmetry  and  in 
poise.  Not  only  are  this  home  and  this  life  thus  rich  in  aesthetic 
interests,  really  controlled  and  sustained  by  motives  of  a  sub- 
limed and  supersensuous  art,  but  the  religious  aspiration,  too, 
finds  there,  especially  in  the  beautiful  doctrine  of  invariants, 
the  most  perfect  symbols  of  what  it  seeks — the  changeless  in 
the  midst  of  change,  abiding  things  in  a  world  of  flux,  con- 
figurations that  remain  the  same  despite  the  swirl  and  stress 
of  countless  hosts  of  curious  transformations.  The  domain  of 
mathematics  is  the  sole  domain  of  certainty.  There  and  there 
alone  prevail  the  standards  by  which  every  hypothesis  respect- 
ing the  external  universe  and  all  observation  and  all  experiment 
must  be  finally  judged.  It  is  the  realm  to  which  all  speculation 
and  all  thought  must  repair  for  chastening  and  sanitation — 
the  court  of  last  resort,  I  say  it  reverently,  for  all  intellection 
whatsoever,  whether  of  demon  or  man  or  deity.  It  is  there  that 
mind  as  mind  attains  its  highest  estate,  and  the  condition  of 
knowledge  there  is  the  ultimate  object,  the  tantalising  goal  of 
the  aspiration,  the  Anders-Streben,  of  all  other  knowledge  of 

every  kind. — KEYSER,  C.  J. 

The  Universe  and  Beyond;  Hibbert  Journal, 
Vol.  3  (1904-1905),  pp.  813-314. 


CHAPTER  III 

ESTIMATES    OF  MATHEMATICS 

301.  The  world  of  ideas  which  it  [mathematics]  discloses  or 
illuminates,   the   contemplation  of  divine  beauty  and  order 
which  it  induces,  the  harmonious  connection  of  its  parts,  the 
infinite  hierarchy  and  absolute  evidence  of  the  truths  with 
which  mathematical  science  is  concerned,  these,  and  such  like, 
are  the  surest  grounds  of  its  title  of  human  regard,  and  would 
remain  unimpaired  were  the  plan  of  the  universe  unrolled  like  a 
map  at  our  feet,  and  the  mind  of  man  qualified  to  take  in  the 
whole  scheme  of  creation  at  a  glance. — SYLVESTER,  J.  J. 

A  Plea  for  the  Mathematician,  Nature,  1,  p. 
262;  Collected  Mathematical  Papers  (Cam- 
bridge, 1908),  2,  p.  659. 

302.  It  may  well  be  doubted  whether,  in  all  the  range  of 
Science,  there  is  any  field  so  fascinating  to  the  explorer — so  rich 
in  hidden  treasures — so  fruitful  in  delightful  surprises — as  that 
of  Pure  Mathematics.     The  charm  lies  chiefly  ...  in  the 
absolute  certainty  of  its  results:  for  that  is  what,  beyond  all 
mental  treasures,  the  human  intellect  craves  for.    Let  us  only 
be  sure  of  something!     More  light,  more  light!    'Ei>  8e  <£aei 
/cat  6\ecrcrov  "  And  if  our  fate  be  death,  give  light  and  let 
us  die!"    This  is  the  cry  that,  through  all  the  ages,  is  going  up 
from  perplexed  Humanity,  and  Science  has  little  else  to  offer, 
that  will  really  meet  the  demands  of  its  votaries,  than  the  con- 
clusions of  Pure  Mathematics. — DODGSON,  C.  L. 

A  New  Theory  of  Parallels  (London,  1895), 
Introduction. 

303.  In  every  case  the  awakening  touch  has  been  the  mathe- 
matical spirit,  the  attempt  to  count,  to  measure,  or  to  calculate. 
What  to  the  poet  or  the  seer  may  appear  to  be  the  very  death  of 
all  his  poetry  and  all  his  visions — the  cold  touch  of  the  cal- 

39 


40  MEMORABILIA  MATHEMATICA 

culating  mind, — this  has  proved  to  be  the  spell  by  which  knowl- 
edge has  been  born,  by  which  new  sciences  have  been  created, 
and  hundreds  of  definite  problems  put  before  the  minds  and 
into  the  hands  of  diligent  students.  It  is  the  geometrical  figure, 
the  dry  algebraical  formula,  which  transforms  the  vague  reason- 
ing of  the  philosopher  into  a  tangible  and  manageable  concep- 
tion; which  represents,  though  it  does  not  fully  describe,  which 
corresponds  to,  though  it  does  not  explain,  the  things  and 
processes  of  nature:  this  clothes  the  fruitful,  but  otherwise 
indefinite,  ideas  in  such  a  form  that  the  strict  logical  methods  of 
thought  can  be  applied,  that  the  human  mind  can  in  its  inner 
chamber  evolve  a  train  of  reasoning  the  result  of  which  corre- 
sponds to  the  phenomena  of  the  outer  world. — MERZ,  J.  T. 
A  History  of  European  Thought  in  the  Nine- 
teenth Century  (Edinburgh  and  London,  1904), 
Vol.  1,  p.  314. 

304.  Mathematics  .  .  .  the  ideal  and  norm  of  all  careful 
thinking. — HALL,  G.  STANLEY. 

Educational  Problems  (New  York,  1911),  p.  893. 

305.  Mathematics  is  the  only  true  metaphysics. 

THOMSON,  W.  (LORD  KELVIN). 
Thompson,  S.  P.:  Life  of  Lord  Kelvin  (London, 
1910),  p.  10. 

306.  He  who  knows  not  mathematics  and  the  results  of  recent 
scientific  investigation  dies  without  knowing  truth. 

SCHELLBACH,    C.    H. 

Quoted  in  Young's  Teaching  of  Mathematics 
(London,  1907),  p.  44- 

307.  The  reasoning  of  mathematics  is  a  type  of  perfect  rea- 
soning.— BARNETT,  P.  A. 

Common  Sense  in  Education  and  Teaching 
(New  York,  1905),  p.  222. 

308.  Mathematics,  once  fairly  established  on  the  foundation 
of  a  few  axioms  and  definitions,  as  upon  a  rock,  has  grown  from 
age  to  age,  so  as  to  become  the  most  solid  fabric  that  human 
reason  can  boast. — REID,  THOMAS. 

Essays  on  the  Intellectual  Powers  of  Man, 
4th.  Ed.,  p.  461. 


ESTIMATES   OF   MATHEMATICS  41 

309.  The  analytical  geometry  of  Descartes  and  the  calculus 
of  Newton  and  Leibniz  have  expanded  into  the  marvelous 
mathematical  method — more  daring  than  anything  that  the 
history  of  philosophy  records — of  Lobachevsky  and  Riemann, 
Gauss  and  Sylvester.  Indeed,  mathematics,  the  indispensable 
tool  of  the  sciences,  defying  the  senses  to  follow  its  splendid 
flights,  is  demonstrating  today,  as  it  never  has  been  demon- 
strated before,  the  supremacy  of  the  pure  reason. 

BUTLER,  NICHOLAS  MURRAY. 

The  Meaning  of  Education  and  other  Essays 
and   Addresses    (New    York,    1905),    p.   45. 


310.  Mathematics  is  the  gate  and  key  of  the  sciences.  .  .  . 
Neglect  of  mathematics  works  injury  to  all  knowledge,  since 
he  who  is  ignorant  of  it  cannot  know  the  other  sciences  or 
the  things  of  this  world.  And  what  is  worse,  men  who  are 
thus  ignorant  are  unable  to  perceive  their  own  ignorance  and 
so  do  not  seek  a  remedy. — BACON,  ROGER. 

Opus  Majus,  Part  4,  Distinctia  Prima,  cap.  1 . 


311.  Just  as  it  will  never  be  successfully  challenged  that  the 
French  language,  progressively  developing  and  growing  more 
perfect  day  by  day,  has  the  better  claim  to  serve  as  a  developed 
court  and  world  language,  so  no  one  will  venture  to  estimate 
lightly  the  debt  which  the  world  owes  to  mathematicians,  in  that 
they  treat  in  their  own  language  matters  of  the  utmost  impor- 
tance, and  govern,  determine  and  decide  whatever  is  subject, 
using  the  word  in  the  highest  sense,  to  number  and  measure- 
ment.— GOETHE . 

Spriiche  in  Prosa,  Natur,  III,  868. 


312.  Do  not  imagine  that  mathematics  is  hard  and  crabbed, 
and  repulsive  to  common  sense.  It  is  merely  the  etherealization 
of  common  sense. — THOMSON,  W.  (LORD  KELVIN). 

Thompson,  S.  P.:  Life  of  Lord  Kelvin  (London, 

1910),  p.  1139. 


42  MEMORABILIA   MATHEMATICA 

313.  The  advancement  and  perfection  of  mathematics  are 
intimately  connected  with  the  prosperity  of  the  State. 

NAPOLEON  I. 

Correspondance  de  Napoleon,  t.  24  (1868), 
p.  112. 

314.  The  love  of  mathematics  is  daily  on  the  increase,  not 
only  with  us  but  in  the  army.    The  result  of  this  was  unmis- 
takably apparent  in  our  last  campaigns.    Bonaparte  himself  has 
a  mathematical  head,  and  though  all  who  study  this  science  may 
not  become  geometricians  like  Laplace  or  Lagrange,  or  heroes 
like  Bonaparte,  there  is  yet  left  an  influence  upon  the  mind 
which  enables  them  to  accomplish  more  than  they  could  possibly 
have  achieved  without  this  training. — LALANDE. 

Quoted  in  Bruhns'  Alexander  von  Humboldt 
(1872),  Bd.  1,  p.  232. 

315.  In  Pure  Mathematics,  where  all  the  various  truths  are 
necessarily  connected  with  each  other,  (being  all  necessarily 
connected  with  those  hypotheses  which  are  the  principles  of  the 
science),   an  arrangement  is  beautiful  in  proportion  as  the 
principles  are  few;  and  what  we  admire  perhaps  chiefly  hi  the 
science,  is  the  astonishing  variety  of  consequences  which  may 
be  demonstrably  deduced  from  so  small  a  number  of  premises. 

STEWART,  DUGALD. 

Philosophy  of  the  Human  Mind,  Part  3, 
chap.  1,  sect.  3;  Collected  Works  [Hamilton] 
(Edinburgh,  1854),  Vol.  4. 

316.  It  is  curious  to  observe  how  differently  these  great  men 
[Plato  and  Bacon]  estimated  the  value  of  every  kind  of  knowl- 
edge.    Take  Arithmetic  for  example.     Plato,  after  speaking 
slightly  of  the  convenience  of  being  able  to  reckon  and  compute 
in  the  ordinary  transactions  of  life,  passes  to  what  he  considers 
as  a  far  more  important  advantage.    The  study  of  the  proper- 
ties of  numbers,  he  tells  us,  habituates  the  mind  to  the  con- 
templation of  pure  truth,  and  raises  us  above  the  material 
universe.    He  would  have  his  disciples  apply  themselves  to  this 
study,  not  that  they  may  be  able  to  buy  or  sell,  not  that  they 
may  qualify  themselves  to  be  shop-keepers  or  travelling  mer- 


ESTIMATES  OF  MATHEMATICS  43 

chants,  but  that  they  may  learn  to  withdraw  their  minds  from 
the  ever-shifting  spectacle  of  this  visible  and  tangible  world, 
and  to  fix  them  on  the  immutable  essences  of  things. 

Bacon,  on  the  other  hand,  valued  this  branch  of  knowledge 
only  on  account  of  its  uses  with  reference  to  that  visible  and 
tangible  world  which  Plato  so  much  despised.  He  speaks  with 
scorn  of  the  mystical  arithmetic  of  the  later  Platonists,  and 
laments  the  propensity  of  mankind  to  employ,  on  mere  matters 
of  curiosity,  powers  the  whole  exertion  of  which  is  required  for 
purposes  of  solid  advantage.  He  advises  arithmeticians  to  leave 
these  trifles,  and  employ  themselves  in  framing  convenient  ex- 
pressions which  may  be  of  use  in  physical  researches. 

MACAULAY. 

Lord  Bacon:  Edinburgh  Review,  July,  1887. 
Critical  and  Miscellaneous  Essays  (New  York, 
1879),  Vol.  1,  p.  897. 

317.  Ath.  There  still  remain  three  studies  suitable  for  free- 
men.    Calculation  in  arithmetic  is  one  of  them;  the  measure- 
ment of  length,  surface,  and  depth  is  the  second;  and  the  third 
has  to  do  with  the  revolutions  of  the  stars  in  reference  to  one 
another  .  .  .    there  is  in  them  something  that  is  necessary  and 
cannot  be  set  aside,  ...  if  I  am  not  mistaken,  [something  of] 
divine  necessity;  for  as  to  the  human  necessities  of  which  men 
often  speak  when  they  talk  in  this  manner,  nothing  can  be  more 
ridiculous  than  such  an  application  of  the  words. 

Cle.  And  what  necessities  of  knowledge  are  there,  Stranger, 
which  are  divine  and  not  human? 

Ath.  I  conceive  them  to  be  those  of  which  he  who  has  no  use 
nor  any  knowledge  at  all  cannot  be  a  god,  or  demi-god,  or  hero 
to  mankind,  or  able  to  take  any  serious  thought  or  charge  of 
them. — PLATO. 

Republic,  Bk.  7.  Jowett's  Dialogues  of  Plato 
(New  York,  1897),  Vol.  4,  P-  834. 

318.  Those  who  assert  that  the  mathematical  sciences  make 
no  affirmation  about  what  is  fair  or  good  make  a  false  assertion; 
for  they  do  speak  of  these  and  frame  demonstrations  of  them  in 
the  most  eminent  sense  of  the  word.    For  if  they  do  not  actually 
employ  these  names,  they  do  not  exhibit  even  the  results  and 


44  MEMORABILIA  MATHEMATICA 

the  reasons  of  these,  and  therefore  can  be  hardly  said  to  make 
any  assertion  about  them.  Of  what  is  fair,  however,  the  most 
important  species  are  order  and  symmetry,  and  that  which  is 
definite,  which  the  mathematical  sciences  make  manifest  in  a 
most  eminent  degree.  And  since,  at  least,  these  appear  to  be 
the  causes  of  many  things — now,  I  mean,  for  example,  order, 
and  that  which  is  a  definite  thing,  it  is  evident  that  they  would 
assert,  also,  the  existence  of  a  cause  of  this  description,  and  its 
subsistence  after  the  same  manner  as  that  which  is  fair  sub- 
sists in. — ARISTOTLE. 

Metaphysics  [MacMahon]  Bk.  12,  chap.  3. 

319.  Many  arts  there  are  which  beautify  the  mind  of  man;  of 
all  other  none  do  more  garnish  and  beautify  it  than  those  arts 
which  are  called  mathematical. — BILLINGSLEY,  H. 

The  Elements  of  Geometric  of  the  most  ancient 
Philosopher  Euclide  of  Megara  (London,  1670}, 
Note  to  the  Reader. 

320.  As  the  sun  eclipses  the  stars  by  his  brilliancy,  so  the 
man  of  knowledge  will  eclipse  the  fame  of  others  in  assemblies  of 
the  people  if  he  proposes  algebraic  problems,  and  still  more  if  he 

solves  them. — BRAHMAGUPTA. 

Quoted  in  Cajori's  History  of  Mathematics 
(New  York,  1897),  p.  92. 

321.  So  highly  did  the  ancients  esteem  the  power  of  figures 
and  numbers,  that  Democritus  ascribed  to  the  figures  of  atoms 
the  first  principles  of  the  variety  of  things;  and  Pythagoras 
asserted  that  the  nature  of  things  consisted  of  numbers. 

BACON,  LORD. 

De  Augmentis,  Bk.  3;  Advancement  of  Learn- 
ing, Bk.  2. 

322.  There  has  not  been  any  science  so  much  esteemed  and 
honored  as  this  of  mathematics,  nor  with  so  much  industry  and 
vigilance  become  the  care  of  great  men,  and  labored  in  by  the 
potentates  of  the  world,  viz.  emperors,  kings,  princes,  etc. 

FRANKLIN,  BENJAMIN. 

On  the  Usefulness  of  Mathematics,  Works 
(Boston,  1840),  Vol.  2,  p.  28. 


ESTIMATES   OF   MATHEMATICS  45 

323.  Whatever  may  have  been  imputed  to  some  other  studies 
under  the  notion  of  insignificancy  and  loss  of  time,  yet  these 
[mathematics],  I  believe,  never  caused  repentance  in  any,  except 
it  was  for  their  remissness  in  the  prosecution  of  them. 

FRANKLIN,  BENJAMIN. 
On   the    Usefulness   of  Mathematics,    Works 
(Boston,  1840),  Vol.  2,  p.  69. 

324.  What  science  can  there  be  more  noble,  more  excellent, 
more  useful  for  men,  more  admirably  high  and  demonstrative, 
than  this  of  the  mathematics? — FRANKLIN,  BENJAMIN. 

On   the    Usefulness   of  Mathematics,    Works 
(Boston,  1840),  Vol.  2,  p.  69. 

325.  The  great  truths  with  which  it  [mathematics]  deals,  are 
clothed  with  austere  grandeur,  far  above  all  purposes  of  im- 
mediate convenience  or  profit.    It  is  in  them  that  our  limited 
understandings  approach  nearest  to  the  conception  of  that  ab- 
solute and  infinite,  towards  which  in  most  other  things  they 
aspire  in  vain.    In  the  pure  mathematics  we  contemplate  abso- 
lute truths,  which  existed  in  the  divine  mind  before  the  morning 
stars  sang  together,  and  which  will  continue  to  exist  there,  when 
the  last  of  their  radiant  host  shall  have  fallen  from  heaven. 
They  existed  not  merely  in  metaphysical  possibility,  but  in  the 
actual  contemplation  of  the  supreme  reason.    The  pen  of  in- 
spiration, ranging  all  nature  and  life  for  imagery  to  set  forth  the 
Creator's  power  and  wisdom,  finds  them  best  symbolized  in  the 
skill  of  the  surveyor.    "He  meted  out  heaven  as  with  a  span;" 
and  an  ancient  sage,  neither  falsely  nor  irreverently,  ventured 
to  say,  that  "God  is  a  geometer." — EVERETT,  EDWARD. 

Orations  and  Speeches  (Boston,  1870),  Vol.  8, 
p.  514. 

326.  There  is  no  science  which  teaches  the  harmonies  of 
nature  more  clearly  than  mathematics,   .   .   . — CARUS,  PAUL. 

Andrews:  Magic  Squares  and  Cubes  (Chicago, 
1908),  Introduction. 

327.  For  it  being  the  nature  of  the  mind  of  man  (to  the 
extreme  prejudice  of  knowledge)  to  delight  in  the  spacious 


46  MEMORABILIA  MATHEMATICA 

liberty  of  generalities,  as  in  a  champion  region,  and  not  in  the 
enclosures  of  particularity;  the  Mathematics  were  the  good- 
liest fields  to  satisfy  that  appetite. — BACON,  LORD. 

De  Augmentis,  Bk.  3;  Advancement  of  Learn- 
ing, Bk.  2. 

328.  I  would  have  my  son  mind  and  understand  business, 
read  little  history,  study  the  mathematics  and  cosmography; 
these  are  good,  with  subordination  to  the  things  of  God.  .  .  . 
These  fit  for  public  services  for  which  man  is  born. 

CROMWELL,  OLIVER. 

Letters  and  Speeches  of  Oliver  Cromwell  (New 
York,  1899),  Vol.  1,  p.  371. 

329.  Mathematics  is  the  life  supreme.    The  life  of  the  gods 
is  mathematics.     All  divine  messengers  are  mathematicians. 
Pure  mathematics  is  religion.    Its  attainment  requires  a  theoph- 
any. — NOVALIS. 

Schriften    (Berlin,    1901),   Bd.   2,   p.   228. 

330.  The  Mathematics  which  effectually  exercises,  not  vainly 
deludes  or  vexatiously  torments  studious  Minds  with  obscure 
Subtilties,  perplexed  Difficulties,  or  contentious  Disquisitions; 
which  overcomes  without  Opposition,  triumphs  without  Pomp, 
compels  without  Force,  and  rules  absolutely  without  Loss  of 
Liberty;  which  does  not  privately  overreach  a  weak  Faith,  but 
openly  assaults  an  armed  Reason,  obtains  a  total  Victory,  and 
puts  on  inevitable  Chains;  whose  Words  are  so  many  Oracles, 
and  Works  as  many  Miracles;  which  blabs  out  nothing  rashly, 
nor  designs  anything  from  the  Purpose,  but  plainly  demonstrates 
and  readily  performs  all  Things  within  its  Verge;  which  ob- 
trudes no  false  Shadow  of  Science,  but  the  very  Science  itself, 
the  Mind  firmly  adheres  to  it,  as  soon  as  possessed  of  it,  and  can 
never  after  desert  it  of  its  own  Accord,  or  be  deprived  of  it  by 
any  Force  of  others:  Lastly  the  Mathematics,  which  depend 
upon  Principles  clear  to  the  Mind,  and  agreeable  to  Experience; 
which  draws  certain  Conclusions,  instructs  by  profitable  Rules, 
unfolds  pleasant  Questions;  and  produces  wonderful  Effects; 
which  is  the  fruitful  Parent  of,  I  had  almost  said  all,  Arts,  the 


ESTIMATES    OF   MATHEMATICS  47 

unshaken  Foundation  of  Sciences,  and  the  plentiful  Fountain 

of  Advantage  to  human  Affairs. — BARROW,  ISAAC. 

Oration  before  the  University  of  Cambridge  on 
being  elected  Lucasian  Professor  of  Mathematics, 
Mathematical  Lectures  (London,  1784},  p.  28. 

331.  Doubtless  the  reasoning  faculty,  the  mind,  is  the  lead- 
ing and  characteristic  attribute  of  the  human  race.  By  the 
exercise  of  this,  man  arrives  at  the  properties  of  the  natural 
bodies.  This  is  science,  properly  and  emphatically  so  called. 
It  is  the  science  of  pure  mathematics;  and  in  the  high  branches 
of  this  science  lies  the  truly  sublime  of  human  acquisition.  If 
any  attainment  deserves  that  epithet,  it  is  the  knowledge, 
which,  from  the  mensuration  of  the  minutest  dust  of  the  bal- 
ance, proceeds  on  the  rising  scale  of  material  bodies,  everywhere 
weighing,  everywhere  measuring,  everywhere  detecting  and  ex- 
plaining the  laws  of  force  and  motion,  penetrating  into  the 
secret  principles  which  hold  the  universe  of  God  together,  and 
balancing  worlds  against  worlds,  and  system  against  system. 
When  we  seek  to  accompany  those  who  pursue  studies  at  once 
so  high,  so  vast,  and  so  exact;  when  we  arrive  at  the  discoveries 
of  Newton,  which  pour  in  day  on  the  works  of  God,  as  if  a 
second  fiat  had  gone  forth  from  his  own  mouth;  when,  further, 
we  attempt  to  follow  those  who  set  out  where  Newton  paused, 
making  his  goal  their  starting-place,  and,  proceeding  with 
demonstration  upon  demonstration,  and  discovery  upon  dis- 
covery, bring  new  worlds  and  new  systems  of  worlds  within 
the  limits  of  the  known  universe,  failing  to  learn  all  only  because 
all  is  infinite;  however  we  may  say  of  man,  in  admiration  of  his 
physical  structure,  that  "in  form  and  moving  he  is  express  and 
admirable,"  it  is  here,  and  here  without  irreverence,  we  may 
exclaim,  "In  apprehension  how  like  a  god!"  The  study  of  the 
pure  mathematics  will  of  course  not  be  extensively  pursued  in 
an  institution,  which,  like  this  [Boston  Mechanics'  Institute], 
has  a  direct  practical  tendency  and  aim*  But  it  is  still  to  be 
remembered,  that  pure  mathematics  lie  at  the  foundation  of 
mechanical  philosophy,  and  that  it  is  ignorance  only  which  can 
speak  or  think  of  that  sublime  science  as  useless  research  or 
barren  speculation. — WEBSTER,  DANIEL. 

Works  (Boston,  1872},  Vol.  1,  p.  180. 


48  MEMORABILIA   MATHEMATICA 

332.  The  school  of  Plato  has  advanced  the  interests  of  the 
race  as  much  through  geometry  as  through  philosophy.    The 
modern  engineer,  the  navigator,  the  astronomer,  built  on  the 
truths  which  those  early  Greeks  discovered  in  their  purely 
speculative  investigations.    And  if  the  poetry,  statesmanship, 
oratory,  and  philosophy  of  our  day  owe  much  to  Plato's  divine 
Dialogues,  our  commerce,  our  manufactures,  and  our  science 
are  equally  indebted  to  his  Conic  Sections.    Later  instances  may 
be  abundantly  quoted,  to  show  that  the  labors  of  the  mathe- 
matician have  outlasted  those  of  the  statesman,  and  wrought 
mightier  changes  in  the  condition  of  the  world.    Not  that  we 
would  rank  the  geometer  above  the  patriot,  but  we  claim  that 
he  is  worthy  of  equal  honor. — HILL,  THOMAS. 

Imagination  in  Mathematics;  North  American 
Review,  Vol.  85,  p.  228. 

333.  The  discoveries  of  Newton  have  done  more  for  England 
and  for  the  race,  than  has  been  done  by  whole  dynasties  of 
British  monarchs;  and  we  doubt  not  that  in  the  great  mathe- 
matical birth  of  1853,  the  Quaternions  of  Hamilton,  there  is  as 
much  real  promise  of  benefit  to  mankind  as  hi  any  event  of 
Victoria's  reign. — HILL,  THOMAS. 

Imagination  in  Mathematics;  North  American 
Review,  Vol.  85,  p.  228. 

334.  Geometrical  and  Mechanical  phenomena  are  the  most 
general,  the  most  simple,  the  most  abstract  of  all, — the  most 
irreducible  to  others.    It  follows  that  the  study  of  them  is  an 
indispensable  preliminary  to  that  of  all  others.     Therefore  must 
Mathematics  hold  the  first  place  in  the  hierarchy  of  the  sciences, 
and  be  the  point  of  departure  of  all  Education,  whether  general 
or  special. — COMTE,  A. 

Positive    Philosophy    [Martineau]    Introduc- 
tion, chap.  2. 


CHAPTER  IV 

THE   VALUE   OF  MATHEMATICS 

401.  Mathematics  because  of  its  nature  and  structure  is 
peculiarly  fitted  for  high  school  instruction  [Gymnasiallehrfach]. 
Especially  the  higher  mathematics,  even  if  presented  only  in  its 
elements,  combines  within  itself  all  those  qualities  which  are 
demanded  of  a  secondary  subject.  It  engages,  it  fructifies,  it 
quickens,  compels  attention,  is  as  circumspect  as  inventive, 
induces  courage  and  self-confidence  as  well  as  modesty  and 
submission  to  truth.  It  yields  the  essence  and  kernel  of  all 
things,  is  brief  in  form  and  overflows  with  its  wealth  of  content. 
It  discloses  the  depth  and  breadth  of  the  law  and  spiritual  ele- 
ment behind  the  surface  of  phenomena;  it  impels  from  point  to 
point  and  carries  within  itself  the  incentive  toward  progress;  it 
stimulates  the  artistic  perception,  good  taste  in  judgment  and 
execution,  as  well  as  the  scientific  comprehension  of  things. 
Mathematics,  therefore,  above  all  other  subjects,  makes  the 
student  lust  after  knowledge,  fills  him,  as  it  were,  with  a  longing 
to  fathom  the  cause  of  things  and  to  employ  his  own  powers 
independently;  it  collects  his  mental  forces  and  concentrates 
them  on  a  single  point  and  thus  awakens  the  spirit  of  individual 
inquiry,  self-confidence  and  the  joy  of  doing;  it  fascinates  be- 
cause of  the  view-points  which  it  offers  and  creates  certainty  and 
assurance,  owing  to  the  universal  validity  of  its  methods.  Thus, 
both  what  he  receives  and  what  he  himself  contributes  toward 
the  proper  conception  and  solution  of  a  problem,  combine  to 
mature  the  student  and  to  make  him  skillful,  to  lead  him  away 
from  the  surface  of  things  and  to  exercise  him  in  the  perception 
of  their  essence.  A  student  thus  prepared  thirsts  after  knowl- 
edge and  is  ready  for  the  university  and  its  sciences.  Thus  it 
appears,  that  higher  mathematics  is  the  best  guide  to  philosophy 
and  to  the  philosophic  conception  of  the  world  (considered  as  a 
self-contained  whole)  and  of  one's  own  being. — DILLMANN,  E. 

Die  Mathematik  die  Fackeltrdgerin  einer  neiten 

Zeit  (Stuttgart,  1889),  p.  40. 

49 


50  MEMORABILIA  MATHEMATICA 

402.  These   Disciplines   [mathematics]   serve  to  inure   and 
corroborate  the  Mind  to  a  constant  Diligence  in  Study;  to 
undergo  the  Trouble  of  an  attentive  Meditation,  and  cheerfully 
contend  with  such  Difficulties  as  lie  in  the  Way.    They  wholly 
deliver  us  from  a  credulous  Simplicity,  most  strongly  fortify  us 
against  the  Vanity  of  Scepticism,  effectually  restrain  from  a 
rash  Presumption,  most  easily  incline  us  to  a  due  Assent,  per- 
fectly subject  us  to  the  Government  of  right  Reason,  and 
inspire  us  with  Resolution  to  wrestle  against  the  unjust  Tyranny 
of  false  Prejudices.    If  the  Fancy  be  unstable  and  fluctuating, 
it  is  to  be  poised  by  this  Ballast,  and  steadied  by  this  Anchor, 
if  the  Wit  be  blunt  it  is  sharpened  upon  this  Whetstone;  if 
luxuriant  it  is  pared  by  this  Knife;  if  headstrong  it  is  restrained 
by  this  Bridle;  and  if  dull  it  is  roused  by  this  Spur.    The  Steps 
are  guided  by  no  Lamp  more  clearly  through  the  dark  Mazes  of 
Nature,  by  no  Thread  more  surely  through  the  intricate  Laby- 
rinths of  Philosophy,  nor  lastly  is  the  Bottom  of  Truth  sounded 
more  happily  by  any  other  Line.    I  will  not  mention  how  plenti- 
ful a  Stock  of  Knowledge  the  Mind  is  furnished  from  these,  with 
what  wholesome  Food  it  is  nourished,  and  what  sincere  Pleasure 
it  enjoys.    But  if  I  speak  farther,  I  shall  neither  be  the  only 
Person,  nor  the  first,  who  affirms  it;  that  while  the  Mind  is 
abstracted  and  elevated  from  sensible  Matter,  distinctly  views 
pure  Forms,  conceives  the  Beauty  of  Ideas,  and  investigates 
the  Harmony  of  Proportions;  the  Manners  themselves  are  sen- 
sibly corrected  and  unproved,  the  Affections  composed  and 
rectified,  the  Fancy  calmed  and  settled,  and  the  Understanding 
raised  and  excited  to  more  divine  Contemplation.    All  which  I 
might  defend  by  Authority,  and  confirm  by  the  Suffrages  of  the 
greatest  Philosophers. — BARROW,  ISAAC. 

Prefatory    Oration:    Mathematical    Lectures 
(London,  1734),  p.  81. 

403.  No  school  subject  so  readily  furnishes  tasks  whose  pur- 
pose can  be  made  so  clear,  so  immediate  and  so  appealing  to  the 
sober  second-thought  of  the  immature  learner  as  the  right  sort 
of  elementary  school  mathematics. — MYERS,  GEORGE. 

Arithmetic    in    Public    School    Education 
(Chicago,  1911),  p.  8. 


THE   VALUE   OF   MATHEMATICS  51 

404.  Mathematics  is  a  type  of  thought  which  seems  ingrained 
in  the  human  mind,  which  manifests  itself  to  some  extent  with 
even  the  primitive  races,  and  which  is  developed  to  a  high  de- 
gree with  the  growth  of  civilization.  ...    A  type  of  thought, 
a  body  of  results,  so  essentially  characteristic  of  the  human 
mind,  so  little  influenced  by  environment,  so  uniformly  present 
in  every  civilization,  is  one  of  which  no  well-informed  mind 
today  can  be  ignorant. — YOUNG,  J.  W.  A. 

The  Teaching  of  Mathematics  (London,  1907), 
p.  14. 

405.  Probably  among  all  the  pursuits  of  the  University, 
mathematics  pre-eminently  demand  self-denial,  patience,  and 
perseverance  from  youth,  precisely  at  that  period  when  they 
have  liberty  to  act  for  themselves,  and  when  on  account  of 
obvious  temptations,   habits  of  restraint  and  application  are 
peculiarly  valuable. — TODHUNTER,  ISAAC. 

The  Conflict  of  Studies  and  other  Essays 
(London,  1873),  p.  12. 

406.  Mathematics  renders  its  best  service  through  the  im- 
mediate furthering  of  rigorous  thought  and  the  spirit  of  inven- 
tion.— HERBAET,  J.  F. 

Mathematischer  Lehrplan  fur  Realschulen: 
Werke  [Kehrbach]  (Langensalza,  1890),  Bd.  5, 
p.  170. 

407.  It  seems  to  me  that  the  older  subjects,  classics  and  math- 
ematics, are  strongly  to  be  recommended  on  the  ground  of  the 
accuracy  with  which  we  can  compare  the  relative  performance 
of  the  students.     In  fact  the  definiteness  of  these  subjects  is 
obvious,  and  is  commonly  admitted.    There  is  however  another 
advantage,  which  I  think  belongs  in  general  to  these  subjects, 
that  the  examinations  can  be  brought  to  bear  on  what  is  really 
most  valuable  in  these  subjects. — TODHUNTER,  ISAAC. 

Conflict  of  Studies  and  other  Essays  (London, 
- 1873),  pp.  6,  7. 

408.  It  is  better  to  teach  the  child  arithmetic  and  Latin 
grammar  than  rhetoric  and  moral  philosophy,  because  they  re- 


52  MEMORABILIA  MATHEMATICA 

quire  exactitude  of  performance  it  is  made  certain  that  the 
lesson  is  mastered,  and  that  power  of  performance  is  worth  more 

than  knowledge. — EMERSON,  R.  W. 

Lecture  on  Education. 


409.  Besides  accustoming  the  student  to  demand  complete 
proof,  and  to  know  when  he  has  not  obtained  it,  mathematical 
studies  are  of  immense  benefit  to  his  education  by  habituating 
him  to  precision.  It  is  one  of  the  peculiar  excellencies  of  mathe- 
matical discipline,  that  the  mathematician  is  never  satisfied  with 
d  pen  pr&s.  He  requires  the  exact  truth.  Hardly  any  of  the 
non-mathematical  sciences,  except  chemistry,  has  this  advan- 
tage. One  of  the  commonest  modes  of  loose  thought,  and 
scources  of  error  both  in  opinion  and  in  practice,  is  to  overlook 
the  importance  of  quantities.  Mathematicians  and  chemists 
are  taught  by  the  whole  course  of  their  studies,  that  the  most 
fundamental  difference  of  quality  depends  on  some  very  slight 
difference  in  proportional  quantity;  and  that  from  the  qualities 
of  the  influencing  elements,  without  careful  attention  to  their 
quantities,  false  expectation  would  constantly  be  formed  as  to 
the  very  nature  and  essential  character  of  the  result  produced. 

MILL,  J.  S. 

An  Examination  of  Sir  William  Hamilton's 
Philosophy  (London,  1878),  p.  611, 


410.  In  mathematics  I  can  report  no  deficience,  except  it  be 
that  men  do  not  sufficiently  understand  the  excellent  use  of 
the  Pure  Mathematics,  in  that  they  do  remedy  and  cure  many 
defects  hi  the  wit  and  faculties  intellectual.  For  if  the  wit  be 
too  dull,  they  sharpen  it;  if  too  wandering,  they  fix  it;  if  too 
inherent  in  the  senses,  they  abstract  it.  So  that  as  tennis  is  a 
game  of  no  use  in  itself,  but  of  great  use  in  respect  it  maketh  a 
quick  eye  and  a  body  ready  to  put  itself  into  all  positions;  so  in 
the  Mathematics,  that  use  which  is  collateral  and  intervenient 
is  no  less  worthy  than  that  which  is  principal  and  intended. 

BACON,  LORD. 

De  Augmentis,  Bk.  3;  Advancement  of  Learn- 
ing, Bk.  2. 


THE   VALUE   OF  MATHEMATICS  53 

411.  If  a  man's  wit  be  wandering,  let  him  study  mathematics; 
for  in  demonstrations,  if  his  wit  be  called  away  never  so  little,  he 

must  begin  again. — BACON,  LORD. 

Essays:  On  Studies. 

412.  If  one  be  bird-witted,  that  is  easily  distracted  and  unable 
to  keep  his  attention  as  long  as  he  should,  mathematics  provides 
a  remedy;  for  in  them  if  the  mind  be  caught  away  but  a  moment, 
the  demonstration  has  to  be  commenced  anew. — BACON,  LORD. 

De  Augmentis,  Bk.  6;  Advancement  of  Learn- 
ing, Bk.  2. 

413.  The  metaphysical  philosopher  from  his  point  of  view 
recognizes  mathematics  as  an  instrument  of  education,  which 
strengthens  the  power  of  attention,  develops  the  sense  of  order 
and  the  faculty  of  construction,  and  enables  the  mind  to  grasp 
under  the  simple  formulae  the  quantitative  differences  of  physi- 
cal phenomena. — JOWETT,  B. 

Dialogues  of  Plato  (New  York,  1897),  Vol.  2, 
p.  78. 

414.  Nor  do  I  know  any  study  which  can  compete  with  math- 
ematics in  general  in  furnishing  matter  for  severe  and  con- 
tinued thought.     Metaphysical  problems  may  be  even  more 
difficult;  but  then  they  are  far  less  definite,  and,  as  they  rarely 
lead  to  any  precise  conclusion,  we  miss  the  power  of  checking  our 
own  operations,  and  of  discovering  whether  we  are  thinking  and 
reasoning  or  merely  fancying  and  dreaming. 

TODHUNTER,    ISAAC. 

Conflict  of  Studies   (London,   1873),  p.   13. 

415.  Another  great  and  special  excellence  of  mathematics  is 
that  it  demands  earnest  voluntary  exertion.     It  is  simply  im- 
possible for  a  person  to  become  a  good  mathematician  by  the 
happy  accident  of  having  been  sent  to  a  good  school;  this  may 
give  him  a  preparation  and  a  start,  but  by  his  own  individual 
efforts  alone  can  he  reach  an  eminent  position. 

TODHUNTER,  ISAAC. 
Conflict  of  Studies    (London,    1873),   p.   2. 


54  MEMORABILIA  MATHEMATICA 

416.  The  faculty  of  resolution  is  possibly  much  invigorated 
by  mathematical  study,  and  especially  by  that  highest  branch 
of  it  which,  unjustly,  merely  on  account  of  its  retrograde  opera- 
tions, has  been  called,  as  if  par  excellence,  analysis. — POE,  E.  A. 

The  Murders  in  Rue  Morgue. 

417.  He  who  gives  a  portion  of  his  time  and  talent  to  the  in- 
vestigation of  mathematical  truth  will  come  to  all  other  ques- 
tions with  a  decided  advantage  over  his  opponents.    He  will  be 
in  argument  what  the  ancient  Romans  were  in  the  field:  to 
them  the  day  of  battle  was  a  day  of  comparative  recreation, 
because  they  were  ever  accustomed  to  exercise  with  arms 
much  heavier  than  they  fought;  and  reviews  differed  from  a  real 
battle  in  two  respects:  they  encountered  more  fatigue,  but  the 

victory  was  bloodless. — COLTON,  C.  C. 

Lacon  (New  York,  1866). 

418.  Mathematics  is  the  study  which  forms  the  foundation 
of  the  course  [West  Point  Military  Academy].    This  is  necessary, 
both  to  impart  to  the  mind  that  combined  strength  and  ver- 
satility, the  peculiar  vigor  and  rapidity  of  comparison  necessary 
for  military  action,  and  to  pave  the  way  for  progress  in  the 
higher  military  sciences. 

Congressional  Committee  on  Military  Affairs, 
1834;  U.  S.  Bureau  of  Education,  Bulletin 
1912,  No.  2,  p.  10. 

419.  Mathematics,  among  all  school  subjects,  is  especially 
adapted  to  further  clearness,  definite  brevity  and  precision  in 
expression,  although  it  offers  no  exercise  in  flights  of  rhetoric. 
This  is  due  in  the  first  place  to  the  logical  rigour  with  which  it 
develops  thought,  avoiding  every  departure  from  the  shortest, 
most  direct  way,  never  allowing  empty  phrases  to  enter.    Other 
subjects  excel  in  the  development  of  expression  in  other  re- 
spects: translation  from  foreign  languages  into  the  mother 
tongue  gives  exercise  in  finding  the  proper  word  for  the  given 
foreign  word  and  gives  knowledge  of  laws  of  syntax,  the  study 
of  poetry  and  prose  furnish  fit  patterns  for  connected  presen- 
tation and  elegant  form  of  expression,  composition  is  to  exercise 
the  pupil  in  a  like  presentation  of  his  own  or  borrowed  thoughts 


THE   VALUE   OF   MATHEMATICS  55 

and  their  development,  the  natural  sciences  teach  description  of 
natural  objects,  apparatus  and  processes,  as  well  as  the  state- 
ment of  laws  on  the  grounds  of  immediate  sense-perception.  But 
all  these  aids  for  exercise  in  the  use  of  the  mother  tongue,  each 
in  its  way  valuable  and  indispensable,  do  not  guarantee,  in  the 
same  manner  as  mathematical  training,  the  exclusion  of  words 
whose  concepts,  if  not  entirely  wanting,  are  not  sufficiently 
clear.  They  do  not  furnish  in  the  same  measure  that  which 
the  mathematician  demands  particularly  as  regards  precision 
of  expression. — REIDT,  F. 

Anleitung  zwn  mathematischen  Unterricht  in 
hoheren  Schukn  (Berlin,  1906),  p.  17. 

420.  One  rarely  hears  of  the  mathematical  recitation  as  a 
preparation  for  public  speaking.  Yet  mathematics  shares  with 
these  studies  [foreign  languages,  drawing  and  natural  science] 
their  advantages,  and  has  another  in  a  higher  degree  than 
either  of  them. 

Most  readers  will  agree  that  a  prime  requisite  for  healthful 
experience  in  public  speaking  is  that  the  attention  of  the  speaker 
and  hearers  alike  be  drawn  wholly  away  from  the  speaker  and 
concentrated  upon  the  thought.  In  perhaps  no  other  class- 
room is  this  so  easy  as  in  the  mathematical,  where  the  close 
reasoning,  the  rigorous  demonstration,  the  tracing  of  neces- 
sary conclusions  from  given  hypotheses,  commands  and  secures 
the  entire  mental  power  of  the  student  who  is  explaining,  and 
of  his  classmates.  In  what  other  circumstances  do  students 
feel  so  instinctively  that  manner  counts  for  so  little  and  mind 
for  so  much?  In  what  other  circumstances,  therefore,  is  a 
simple,  unaffected,  easy,  graceful  manner  so  naturally  and  so 
healthfully  cultivated?  Mannerisms  that  are  mere  affectation 
or  the  result  of  bad  literary  habit  recede  to  the  background  and 
finally  disappear,  while  those  peculiarities  that  are  the  expres- 
sion of  personality  and  are  inseparable  from  its  activity  con- 
tinually develop,  where  the  student  frequently  presents,  to  an 
audience  of  his  intellectual  peers,  a  connected  train  of  reason- 
ing. .  .  . 

One  would  almost  wish  that  our  institutions  of  the  science 
and  art  of  public  speaking  would  put  over  their  doors  the  motto 


56  MEMORABILIA   MATHEMATICA 

that  Plato  had  over  the  entrance  to  his  school  of  philosophy: 
"Let  no  one  who  is  unacquainted  with  geometry  enter  here." 

WHITE,  W.  F. 

A    Scrap-book    of   Elementary    Mathematics 
(Chicago,  1908),  p.  210. 

421.  The  training  which  mathematics  gives  in  working  with 
symbols  is  an  excellent  preparation  for  other  sciences;  .  .  .  the 
world's  work  requires  constant  mastery  of  symbols. 

YOUNG,  J.  W.  A. 

The   Teaching  of  Mathematics   (New   York, 
1907),  p.  1$. 

422.  One  striking  peculiarity  of  mathematics  is  its  unlimited 
power  of  evolving  examples  and  problems.    A  student  may  read 
a  book  of  Euclid,  or  a  few  chapters  of  Algebra,  and  within  that 
limited  range  of  knowledge  it  is  possible  to  set  him  exercises  as 
real  and  as  interesting  as  the  propositions  themselves  which  he 
has  studied;  deductions  which  might  have  pleased  the  Greek 
geometers,  and  algebraic  propositions  which  Pascal  and  Fermat 
would  not  have  disdained  to  investigate. — TODHUNTER,  ISAAC. 

Private   Study   of  Mathematics:    Conflict   of 
Studies  and  other  Essays  (London,  1878),  p.  82. 

423.  Would  you  have  a  man  reason  well,  you  must  use  him 
to  it  betimes;  exercise  his  mind  in  observing  the  connection 
between  ideas,  and  following  them  in  train.    Nothing  does  this 
better  than  mathematics,  which  therefore,  I  think  should  be 
taught  to  all  who  have  the  time  and  opportunity,  not  so  much 
to  make  them  mathematicians,  as  to  make  them  reasonable 
creatures;  for  though  we  all  call  ourselves  so,  because  we  are 
born  to  it  if  we  please,  yet  we  may  truly  say  that  nature  gives 
us  but  the  seeds  of  it,  and  we  are  carried  no  farther  than  indus- 
try and  application  have  carried  us. — LOCKE,  JOHN. 

Conduct  of  the  Understanding,  Sect.  6. 

424.  Secondly,  the  study  of  mathematics  would  show  them 
the  necessity  there  is  in  reasoning,  to  separate  all  the  distinct 
ideas,  and  to  see  the  habitudes  that  all  those  concerned  in  the 
present  inquiry  have  to  one  another,  and  to  lay  by  those  which 
relate  not  to  the  proposition  in  hand,  and  wholly  to  leave  them 


THE   VALUE   OF   MATHEMATICS  57 

out  of  the  reckoning.  This  is  that  which,  in  other  respects 
besides  quantity  is  absolutely  requisite  to  just  reasoning,  though 
in  them  it  is  not  so  easily  observed  and  so  carefully  practised. 
In  those  parts  of  knowledge  where  it  is  thought  demonstration 
has  nothing  to  do,  men  reason  as  it  were  in  a  lump;  and  if  upon  a 
summary  and  confused  view,  or  upon  a  partial  consideration, 
they  can  raise  the  appearance  of  a  probability,  they  usually  rest 
content;  especially  if  it  be  in  a  dispute  where  every  little  straw 
is  laid  hold  on,  and  everything  that  can  but  be  drawn  in  any 
way  to  give  color  to  the  argument  is  advanced  with  ostentation. 
But  that  mind  is  not  in  a  posture  to  find  truth  that  does  not 
distinctly  take  all  the  parts  asunder,  and,  omitting  what  is  not 
at  all  to  the  point,  draws  a  conclusion  from  the  result  of  all  the 
particulars  which  in  any  way  influence  it. — LOCKE,  JOHN. 

Conduct  of  the  Understanding,  Sect.  7. 

426.  I  have  before  mentioned  mathematics,  wherein  algebra 
gives  new  helps  and  views  to  the  understanding.  If  I  propose 
these  it  is  not  to  make  every  man  a  thorough  mathematician 
or  deep  algebraist;  but  yet  I  think  the  study  of  them  is  of 
infinite  use  even  to  grown  men;  first  by  experimentally  convinc- 
ing them,  that  to  make  anyone  reason  well,  it  is  not  enough  to 
have  parts  wherewith  he  is  satisfied,  and  that  serve  him  well 
enough  in  his  ordinary  course.  A  man  in  those  studies  will  see, 
that  however  good  he  may  think  his  understanding,  yet  in 
many  things,  and  those  very  visible,  it  may  fail  him.  This  would 
take  off  that  presumption  that  most  men  have  of  themselves  in 
this  part;  and  they  would  not  be  so  apt  to  think  their  minds 
wanted  no  helps  to  enlarge  them,  that  there  could  be  nothing 
added  to  the  acuteness  and  penetration  of  their  understanding. 

LOCKE,  JOHN. 
The  Conduct  of  the  Understanding,  Sect.  7. 

426.  I  have  mentioned  mathematics  as  a  way  to  settle  in  the 
mind  a  habit  of  reasoning  closely  and  in  train;  not  that  I  think 
it  necessary  that  all  men  should  be  deep  mathematicians,  but 
that,  having  got  the  way  of  reasoning  which  that  study  neces- 
sarily brings  the  mind  to,  they  might  be  able  to  transfer  it  to 
other  parts  of  knowledge,  as  they  shall  have  occasion.  For  in 


58  MEMORABILIA   MATHEMATICA 

all  sorts  of  reasoning,  every  single  argument  should  be  managed 
as  a  mathematical  demonstration;  the  connection  and  depend- 
ence of  ideas  should  be  followed  till  the  mind  is  brought  to  the 
source  on  which  it  bottoms,  and  observes  the  coherence  all 
along;  .  .  .  . — LOCKE,  JOHN. 

The  Conduct  of  the  Understanding,  Sect.  7. 

427.  As  an  exercise  of  the  reasoning  faculty,  pure  mathe- 
matics is  an  admirable  exercise,  because  it  consists  of  reasoning 
alone,  and  does  not  encumber  the  student  with  an  exercise  of 
judgment:  and  it  is  well  to  begin  with  learning  one  thing  at  a 
time,  and  to  defer  a  combination  of  mental  exercises  to  a  later 
period. — WHATELY,  R. 

Annotations  to  Bacon's  Essays  (Boston,  1873), 
Essay  1,  p.  493. 

428.  It  hath  been  an  old  remark,  that  Geometry  is  an  excel- 
lent Logic.    And  it  must  be  owned  that  when  the  definitions  are 
clear;  when  the  postulata  cannot  be  refused,  nor  the  axioms 
denied;  when  from  the  distinct  contemplation  and  comparison 
of  figures,  their  properties  are  derived,  by  a  perpetual  well- 
connected  chain  of  consequences,  the  objects  being  still  kept  in 
view,  and  the  attention  ever  fixed  upon  them;  there  is  acquired  a 
habit  of  reasoning,  close  and  exact  and  methodical ;  which  habit 
strengthens  and  sharpens  the  mind,  and  being  transferred  to 
other  subjects  is  of  general  use  in  the  inquiry  after  truth. 

BERKELY,  GEORGE. 

The  Analyst,  2;  Works  (London,  1898),  Vol. 
3,  p.  10. 

429.  Suppose  then  I  want  to  give  myself  a  little  training  in  the 
art  of  reasoning;  suppose  I  want  to  get  out  of  the  region  of  con- 
jecture and  probability,  free  myself  from  the  difficult  task  of 
weighing  evidence,  and  putting  instances  together  to  arrive  at 
general  propositions,  and  simply  desire  to  know  how  to  deal 
with  my  general  propositions  when  I  get  them,  and  how  to 
deduce  right  inferences  from  them;  it  is  clear  that  I  shall  obtain 
this  sort  of  discipline  best  in  those  departments  of  thought  in 
which  the  first  principles  are  unquestionably  true.    For  in  all 


THE   VALUE    OF  MATHEMATICS  59 

our  thinking,  if  we  come  to  erroneous  conclusions,  we  come  to 
them  either  by  accepting  false  premises  to  start  with — in  which 
case  our  reasoning,  however  good,  will  not  save  us  from  error; 
or  by  reasoning  badly,  in  which  case  the  data  we  start  from  may 
be  perfectly  sound,  and  yet  our  conclusions  may  be  false.  But 
in  the  mathematical  or  pure  sciences, — geometry,  arithmetic, 
algebra,  trigonometry,  the  calculus  of  variations  or  of  curves, — 
we  know  at  least  that  there  is  not,  and  cannot  be,  error  in  our 
first  principles,  and  we  may  therefore  fasten  our  whole  attention 
upon  the  processes.  As  mere  exercises  in  logic,  therefore,  these 
sciences,  based  as  they  all  are  on  primary  truths  relating  to 
space  and  number,  have  always  been  supposed  to  furnish  the 
most  exact  discipline.  When  Plato  wrote  over  the  portal  of  his 
school.  "Let  no  one  ignorant  of  geometry  enter  here/'  he  did  not 
mean  that  questions  relating  to  lines  and  surfaces  would  be 
discussed  by  his  disciples.  On  the  contrary,  the  topics  to  which 
he  directed  their  attention  were  some  of  the  deepest  problems, — 
social,  political,  moral, — on  which  the  mind  could  exercise  itself. 
Plato  and  his  followers  tried  to  think  out  together  conclusions 
respecting  the  being,  the  duty,  and  the  destiny  of  man,  and  the 
relation  in  which  he  stood  to  the  gods  and  to  the  unseen  world. 
What  had  geometry  to  do  with  these  things?  Simply  this: 
That  a  man  whose  mind  has  not  undergone  a  rigorous  training 
in  systematic  thinking,  and  in  the  art  of  drawing  legitimate 
inferences  from  premises,  was  unfitted  to  enter  on  the  discussion 
of  these  high  topics;  and  that  the  sort  of  logical  discipline  which 
he  needed  was  most  likely  to  be  obtained  from  geometry — the 
only  mathematical  science  which  in  Plato's  time  had  been  for- 
mulated and  reduced  to  a  system.  And  we  in  this  country 
[England]  have  long  acted  on  the  same  principle.  Our  future 
lawyers,  clergy,  and  statesmen  are  expected  at  the  University 
to  learn  a  good  deal  about  curves,  and  angles,  and  numbers  and 
proportions;  not  because  these  subjects  have  the  smallest  re- 
lation to  the  needs  of  their  lives,  but  because  in  the  very  act  of 
learning  them  they  are  likely  to  acquire  that  habit  of  steadfast 
and  accurate  thinking,  which  is  indispensable  to  success  in  all 
the  pursuits  of  life. — FITCH,  J.  C. 

Lectures    on    Teaching    (New    York,    1906), 
pp.  291-292. 


60  MEMORABILIA  MATHEMATICA 

430.  It  is  admitted  by  all  that  a  finished  or  even  a  competent 
reasoner  is  not  the  work  of  nature  alone;  the  experience  of 
every  day  makes  it  evident  that  education  develops  faculties 
which  would  otherwise  never  have  manifested  their  existence. 
It  is,  therefore,  as  necessary  to  learn  to  reason  before  we  can 
expect  to  be  able  to  reason,  as  it  is  to  learn  to  swim  or  fence,  in 
order  to  attain  either  of  those  arts.  Now,  something  must  be 
reasoned  upon,  it  matters  not  much  what  it  is,  provided  it  can 
be  reasoned  upon  with  certainty.  The  properties  of  mind  or 
matter,  or  the  study  of  languages,  mathematics,  or  natural 
history,  may  be  chosen  for  this  purpose.  Now  of  all  these,  it  is 
desirable  to  choose  the  one  which  admits  of  the  reasoning  being 
verified,  that  is,  in  which  we  can  find  out  by  other  means,  such 
as  measurement  and  ocular  demonstration  of  all  sorts,  whether 
the  results  are  true  or  not.  When  the  guiding  property  of  the 
loadstone  was  first  ascertained,  and  it  was  necessary  to  learn 
how  to  use  this  new  discovery,  and  to  find  out  how  far  it  might 
be  relied  on,  it  would  have  been  thought  advisable  to  make  many 
passages  between  ports  that  were  well  known  before  attempting 
a  voyage  of  discovery.  So  it  is  with  our  reasoning  faculties :  it  is 
desirable  that  their  powers  should  be  exerted  upon  objects  of 
such  a  nature,  that  we  can  tell  by  other  means  whether  the 
results  which  we  obtain  are  true  or  false,  and  this  before  it  is 
safe  to  trust  entirely  to  reason.  Now  the  mathematics  are 
peculiarly  well  adapted  for  this  purpose,  on  the  following 
grounds: 

1.  Every  term  is  distinctly  explained,  and  has  but  one  mean- 
ing, and  it  is  rarely  that  two  words  are  employed  to  mean  the 
same  thing. 

2.  The  first  principles  are  self-evident,  and,  though  derived 
from  observation,  do  not  require  more  of  it  than  has  been  made 
by  children  in  general. 

3.  The  demonstration  is  strictly  logical,  taking  nothing  for 
granted   except   self-evident   first   principles,   resting  nothing 
upon  probability,  and  entirely  independent  of  authority  and 
opinion. 

4.  When  the  conclusion  is  obtained  by  reasoning,  its  truth  or 
falsehood  can  be  ascertained,  in  geometry  by  actual  measure- 
ment, in  algebra  by  common  arithmetical  calculation.     This 


THE   VALUE   OF  MATHEMATICS  61 

gives  confidence,  and  is  absolutely  necessary,  if,  as  was  said 
before,  reason  is  not  to  be  the  instructor,  but  the  pupil. 

6.  There  are  no  words  whose  meanings  are  so  much  alike  that 
the  ideas  which  they  stand  for  may  be  confounded.  Between 
the  meaning  of  terms  there  is  no  distinction,  except  a  total 
distinction,  and  all  adjectives  and  adverbs  expressing  difference 
of  degrees  are  avoided. — DE  MORGAN,  AUGUSTUS. 

On  the  Study  and  Difficulties  of  Mathematics 
(Chicago,  1898),  chap.  1. 

431.  The  instruction  of  children  should  aim  gradually  to 
combine  knowing  and  doing  [Wissen  und  Konnen].     Among  all 
sciences  mathematics  seems  to  be  the  only  one  of  a  kind  to 
satisfy  this  aim  most  completely. — KANT,  IMMANUEL. 

Werke  [Rosenkranz  und  Schubert],  Bd.  9 
(Leipzig,  1838),  p.  409. 

432.  Every  discipline  must  be  honored  for  reason  other  than 
its  utility,  otherwise  it  yields  no  enthusiasm  for  industry. 

For  both  reasons,  I  consider  mathematics  the  chief  subject 
for  the  common  school.  No  more  highly  honored  exercise 
for  the  mind  can  be  found;  the  buoyancy  [Spannkraft]  which 
it  produces  is  even  greater  than  that  produced  by  the  an- 
cient languages,  while  its  utility  is  unquestioned. 

HERBART,  J.  F. 

Mathematischer  Lehrplan  fur  Realgymnasien, 
Werke  [Kehrbach],  (Langensalza,  1890),  Bd.  5, 
p.  167. 

433.  The  motive  for  the  study  of  mathematics  is  insight  into 
the  nature  of  the  universe.     Stars  and  strata,  heat  and  elec- 
tricity, the  laws  and  processes  of  becoming  and  being,  incorpo- 
rate mathematical  truths.     If  language  imitates  the  voice  of 
the  Creator,  revealing  His  heart,  mathematics  discloses  His 
intellect,  repeating  the  story  of  how  things  came  into  being. 
And  the  value  of  mathematics,  appealing  as  it  does  to  our 
energy  and  to  our  honor,  to  our  desire  to  know  the  truth  and 
thereby  to  live  as  of  right  in  the  household  of  God,  is  that  it 
establishes  us  in  larger  and  larger  certainties.     As  literature 


62  MEMORABILIA   MATHEMATICA 

develops  emotion,  understanding,  and  sympathy,  so  mathe- 
matics develops  observation,  imagination,  and  reason. 

CHANCELLOR,  W.  E. 
A  Theory  of  Motives,  Ideals  and  Values  in 
Education  (Boston    and  New    York,    1907), 
p.  406. 

434.  Mathematics  in  its  pure  form,  as  arithmetic,  algebra, 
geometry,  and  the  applications  of  the  analytic  method,  as  well 
as  mathematics  applied  to  matter  and  force,  or  statics  and 
dynamics,  furnishes  the  peculiar  study  that  gives  to  us,  whether 
as  children  or  as  men,  the  command  of  nature  in  this  its  quantita- 
tive aspect;  mathematics  furnishes  the  instrument,  the  tool  of 
thought,  which  we  wield  in  this  realm. — HARRIS,  W.  T. 

Psychologic  Foundations  of  Education  (New 
York,  1898),  p.  325. 

435.  Little  can  be  understood  of  even  the  simplest  phenomena 
of  nature  without  some  knowledge  of  mathematics,  and  the 
attempt  to  penetrate  deeper  into  the  mysteries  of  nature  compels 
simultaneous  development  of  the  mathematical  processes. 

YOUNG,  J.  W.  A. 

The  Teaching  of  Mathematics  (New  York, 
1907),  p.  16. 

436.  For  many  parts  of  nature  can  neither  be  invented  with 
sufficient  subtility  nor  demonstrated  with  sufficient  perspicuity 
nor  accommodated  unto  use  with  sufficient  dexterity,  without 
the  aid  and  intervening  of  mathematics. — BACON,  LORD. 

De  Augmentis,  Bk.  2;  Advancement  of  Learn- 
ing, Bk.  8. 

437.  I  confess,  that  after  I  began  ...  to  discern  how  useful 
mathematicks  may  be  made  to  physicks,  I  have  often  wished 
that  I  had  employed  about  the  speculative  part  of  geometry, 
and  the  cultivation  of  the  specious  Algebra  I  had  been  taught 
very  young,  a  good  part  of  that  time  and  industry,  that  I  had 
spent  about  surveying  and  fortification  (of  which  I  remember 
I  once  wrote  an  entire  treatise)  and  other  parts  of  practick 
mathematicks. — BOYLE,  ROBERT. 

The  Usefulness  of  Mathematiks  to  Natural 
Philosophy;  Works  (London,  1772),  Vol.  3, 
p.  426. 


THE   VALUE    OF   MATHEMATICS  63 

438.  Mathematics  gives   the  young  man  a   clear  idea  of 
demonstration   and  habituates   him   to   form   long  trains   of 
thought  and  reasoning  methodically  connected  and  sustained 
by  the  final  certainty  of  the  result;  and  it  has  the  further  advan- 
tage, from  a  purely  moral  point  of  view,  of  inspiring  an  absolute 
and  fanatical  respect  for  truth.   In  addition  to  all  this,  mathe- 
matics, and  chiefly  algebra  and  infinitesimal  calculus,  excite  to  a 
high  degree  the  conception  of  the  signs  and  symbols — necessary 
instruments  to  extend  the  power  and  reach  of  the  human  mind 
by  summarizing  an  aggregate  of  relations  in  a  condensed  form 
and  in  a  kind  of  mechanical  way.    These  auxiliaries  are  of  spe- 
cial value  in  mathematics  because  they  are  there  adequate 
to  their  definitions,  a  characteristic  which  they  do  not  possess  to 
the  same  degree  in  the  physical  and  mathematical  [natural?] 
sciences. 

There  are,  in  fact,  a  mass  of  mental  and  moral  faculties  that 
can  be  put  in  full  play  only  by  instruction  in  mathematics;  and 
they  would  be  made  still  more  available  if  the  teaching  was 
directed  so  as  to  leave  free  play  to  the  personal  work  of  the 
student. — BERTHELOT,  M.  P.  E.  M. 

Science  as  an  Instrument  of  Education;  Popu- 
lar Science  Monthly  (1897},  p.  253. 

439.  Mathematical  knowledge,  therefore,  appears  to  us  of 
value  not  only  in  so  far  as  it  serves  as  means  to  other  ends,  but 
for  its  own  sake  as  well,  and  we  behold,  both  in  its  systematic 
external  and  internal  development,  the  most  complete  and 
purest  logical  mind-activity,  the  embodiment  of  the  highest 

intellect-esthetics. — PRINGSHEIM,  ALFRED. 

Ueber  Wert  und  angeblichen  Unwert  der  Mathe- 
matik;  Jahresbericht  der  Deutschen  Mathe- 
matiker  Vereinigung,  Bd.  13,  p.  381. 

440.  The  advantages  which  mathematics  derives  from  the 
peculiar  nature  of  those  relations  about  which  it  is  conversant, 
from  its  simple  and  definite  phraseology,  and  from  the  severe 
logic  so  admirably  displayed  in  the  concatenation  of  its  innum- 
erable theorems,  are  indeed  immense,  and  well  entitled  to  sep- 
arate and  ample  illustration. — STEWART,  DUGALD. 

Philosophy  of    the    Human  Mind,  Part  8, 
chap.  2,  sect.  3. 


64  MEMORABILIA   MATHEMATICA 

441.  I  do  not  intend  to  go  deeply  into  the  question  how  far 
mathematical  studies,  as  the  representatives  of  conscious  log- 
ical reasoning,  should  take  a  more  important  place  in  school 
education.    But  it  is,  in  reality,  one  of  the  questions  of  the  day. 
In  proportion  as  the  range  of  science  extends,  its  system  and 
organization  must  be  improved,  and  it  must  inevitably  come 
about  that  individual  students  will  find  themselves  compelled 
to  go  through  a  stricter  course  of  training  than  grammar  is  in  a 
position  to  supply.    What  strikes  me  in  my  own  experience  with 
students  who  pass  from  our  classical  schools  to  scientific  and 
medical  studies,  is  first,  a  certain  laxity  in  the  application  of 
strictly  universal  laws.    The  grammatical  rules,  in  which  they 
have  been  exercised,  are  for  the  most  part  followed  by  long  lists 
of  exceptions;  accordingly  they  are  not  in  the  habit  of  relying 
implicitly  on  the  certainty  of  a  legitimate  deduction  from  a 
strictly  universal  law.    Secondly,  I  find  them  for  the  most  part 
too  much  inclined  to  trust  to  authority,  even  in  cases  where 
they  might  form  an  independent  judgment.    In  fact,  in  philolog- 
ical studies,  inasmuch  as  it  is  seldom  possible  to  take  in  the 
whole  of  the  premises  at  a  glance,  and  inasmuch  as  the  decision 
of  disputed  questions  often  depends  on  an  aesthetic  feeling  for 
beauty  of  expression,  or  for  the  genius  of  the  language,  attain- 
able only  by  long  training,  it  must  often  happen  that  the  student 
is  referred  to  authorities  even  by  the  best  teachers.    Both  faults 
are  traceable  to  certain  indolence  and  vagueness  of  thought, 
the  sad  effects  of  which  are  not  confined  to  subsequent  scientific 
studies.    But  certainly  the  best  remedy  for  both  is  to  be  found 
in  mathematics,  where  there  is  absolute  certainty  in  the  reason- 
ing, and  no  authority  is  recognized  but  that  of  one's  own  intelli- 
gence.— HELMHOLTZ,  H. 

On  the  Relation  of  Natural  Science  to  Science  in 
general;  Popular  Lectures  on  Scientific  Sub- 
jects; Atkinson  (New  York',  1900),  pp.  25-26. 

442.  What  renders  a  problem  definite,  and  what  leaves  it 
indefinite,  may  best  be  understood  from  mathematics.     The 
very  important  idea  of  solving  a  problem  within  limits  of  error 
is  an  element  of  rational  culture,  coming  from  the  same  source. 
The  art  of  totalizing  fluctuations  by  curves  is  capable  of  being 
carried,  in  conception,  far  beyond  the  mathematical  domain, 


THE   VALUE   OF   MATHEMATICS  65 

where  it  is  first  learned.  The  distinction  between  laws  and 
coefficients  applies  in  every  department  of  causation.  The 
theory  of  Probable  Evidence  is  the  mathematical  contribution 

to  Logic,  and  is  of  paramount  importance. — BAIN,  ALEXANDER. 

Education  as  a  Science  (New  York,  1898), 
pp.  151-152. 

443.  We  receive  it  as  a  fact,  that  some  minds  are  so  con- 
stituted as  absolutely  to  require  for  their  nurture  the  severe 
logic  of  the  abstract  sciences;  that  rigorous  sequence  of  ideas 
which  leads  from  the  premises  to  the  conclusion,  by  a  path, 
arduous  and  narrow,  it  may  be,  and  which  the  youthful  reason 
may  find  it  hard  to  mount,  but  where  it  cannot  stray;  and  on 
which,  if  it  move  at  all,  it  must  move  onward  and  upward.  .  .  . 
Even  for  intellects  of  a  different  character,   whose  natural 
aptitude  is  for  moral  evidence  and  those  relations  of  ideas 
which  are  perceived  and  appreciated  by  taste,  the  study  of  the 
exact  sciences  may  be  recommended  as  the  best  protection 
against  the  errors  into  which  they  are  most  likely  to  fall.    Al- 
though the  study  of  language  is  in  many  respects  no  mean 
exercise  in  logic,  yet  it  must  be  admitted  that  an  eminently 
practical  mind  is  hardly  to  be  formed  without  mathematical 
training. — EVERETT,  EDWARD. 

Orations  and  Speeches  (Boston,  1870),  Vol.  2, 
p.  510. 

444.  The  value  of  mathematical  instruction  as  a  preparation 
for  those  more  difficult  investigations,  consists  in  the  applica- 
bility not  of  its  doctrines  but  of  its  methods.     Mathematics 
will  ever  remain  the  past  perfect  type  of  the  deductive  method 
in  general;  and  the  applications  of  mathematics  to  the  simpler 
branches  of  physics  furnish  the  only  school  in  which  philosophers 
can  effectually  learn  the  most  difficult  and  important  of  their 
art,  the  employment  of  the  laws  of  simpler  phenomena  for  ex- 
plaining and  predicting  those  of  the  more  complex.    These 
grounds  are  quite  sufficient  for  deeming  mathematical  training 
an  indispensable  basis  of  real  scientific  education,  and  regarding 
with  Plato,  one  who  is  dyeco^erprjTo^,  as  wanting  in  one  of  the 
most  essential  qualifications  for  the  successful  cultivation  of  the 
higher  branches  of  philosophy — MILL,  J.  S. 

System  of  Logic,  Bk.  3,  chap.  24,  sect.  9. 


66  MEMORABILIA  MATHEMATICA 

445.  This  science,  Geometry,  is  one  of  indispensable  use  and 
constant  reference,  for  every  student  of  the  laws  of  nature;  for 
the  relations  of  space  and  number  are  the  alphabet  in  which 
those  laws  are  written.    But  besides  the  interest  and  importance 
of  this  kind  which  geometry  possesses,  it  has  a  great  and 
peculiar  value  for  all  who  wish  to  understand  the  foundations  of 
human  knowledge,  and  the  methods  by  which  it  is  acquired. 
For  the  student  of  geometry  acquires,  with  a  degree  of  insight 
and  clearness  which  the  unmathematical  reader  can  but  feebly 
imagine,  a  conviction  that  there  are  necessary  truths,  many  of 
them  of  a  very  complex  and  striking  character;  and  that  a  few 
of  the  most  simple  and  self-evident  truths  which  it  is  possible 
for  the  mind  of  man  to  apprehend,  may,  by  systematic  de- 
duction, lead  to  the  most  remote  and  unexpected  results. 

WHEWELL,  WILLIAM. 
The   Philosophy   of  the  Inductive    Sciences, 
Part  1,  Bk.  2,  chap.  4,  sect.  8  (London,  1858). 

446.  Mathematics,  while  giving  no  quick  remuneration,  like 
the  art  of  stenography  or  the  craft  of  bricklaying,  does  furnish 
the  power  for  deliberate  thought  and  accurate  statement,  and 
to  speak  the  truth  is  one  of  the  most  social  qualities  a  person 
can  possess.    Gossip,  flattery,  slander,  deceit,  all  spring  from  a 
slovenly  mind  that  has  not  been  trained  in  the  power  of  truthful 
statement,  which  is  one  of  the  highest  utilities. — BUTTON,  S.  T. 

Social  Phases  of  Education  in  the  School  and 
the  Home  (London,  1900),  p.  80. 

447.  It  is  from  this  absolute  indifference  and  tranquility  of 
the  mind,  that  mathematical  speculations  derive  some  of  their 
most  considerable  advantages;  because  there  is  nothing  to 
interest  the  imagination;  because  the  judgment  sits  free  and 
unbiased  to  examine  the  point.    All  proportions,  every  arrange- 
ment of  quantity,  is  alike  to  the  understanding,  because  the 
same  truths  result  to  it  from  all;  from  greater  from  lesser,  from 
equality  and  inequality. — BURKE,  EDMUND. 

On  the  Sublime  and  Beautiful,  Part  3,  sect.  2. 

448.  Out  of  the  interaction  of  form  and  content  in  mathe- 
matics grows  an  acquaintance  with  methods  which  enable  the 


THE   VALUE   OF  MATHEMATICS  67 

student  to  produce  independently  within  certain  though  mod- 
erate limits,  and  to  extend  his  knowledge  through  his  own  re- 
flection. The  deepening  of  the  consciousness  of  the  intellectual 
powers  connected  with  this  kind  of  activity,  and  the  gradual 
awakening  of  the  feeling  of  intellectual  self-reliance  may  well  be 
considered  as  the  most  beautiful  and  highest  result  of  mathe- 
matical training. — PRINGSHEIM,  ALFRED. 

Ueber  Wert  und  angeblichen  Unwert  der  Mathe- 
matik;  Jahresbericht  der  Deutschen  Mathe- 
matiker  Vereinigung  (1904),  P-  374- 

449.  He  who  would  know  what  geometry  is,  must  venture 
boldly  into  its  depths  and  learn  to  think  and  feel  as  a  geometer. 
I  believe  that  it  is  impossible  to  do  this,  and  to  study  geometry 
as  it  admits  of  being  studied  and  am  conscious  it  can  be  taught, 
without  finding  the  reason  invigorated,  the  invention  quickened, 
the  sentiment  of  the  orderly  and  beautiful  awakened  and  en- 
hanced, and  reverence  for  truth,  the  foundation  of  all  integrity 
of  character,  converted  into  a  fixed  principle  of  the  mental  and 
moral  constitution,  according  to  the  old  and  expressive  adage 
"abeunt  studio,  in  mores." — SYLVESTER,  J.  J. 

A  probationary  Lecture  on  Geometry;  Collected 
Mathematical  Papers  (Cambridge,  1908),  Vol. 
2,  p.  9. 

450.  Mathematical  knowledge  adds  vigour  to  the  mind,  frees 
it  from  prejudice,  credulity,  and  superstition. 

ARBUTHNOT,  JOHN. 
Usefulness    of    Mathematical    Learning. 

451.  When  the  boy  begins  to  understand  that  the  visible 
point  is  preceded  by  an  invisible  point,  that  the  shortest  dis- 
tance between  two  points  is  conceived  as  a  straight  line  before 
it  is  ever  drawn  with  the  pencil  on  paper,  he  experiences  a  feeling 
of  pride,  of  satisfaction.    And  justly  so,  for  the  fountain  of  all 
thought  has  been  opened  to  him,  the  difference  between  the 
ideal  and  the  real,  potentia  et  actu,  has  become  clear  to  him; 
henceforth  the  philosopher  can  reveal  him  nothing  new,  as  a 
geometrician  he  has  discovered  the  basis  of  all  thought. 

GOETHE. 
Spriiche  in  Prosa,  Ethisches,  VI,  455. 


68  MEMORABILIA   MATHEMATICA 

462.  In  mathematics,  .  .  .  and  in  natural  philosophy  since 
mathematics  was  applied  to  it,  we  see  the  noblest  instance  of 
the  force  of  the  human  mind,  and  of  the  sublime  heights  to 
which  it  may  rise  by  cultivation.    An  acquaintance  with  such 
sciences  naturally  leads  us  to  think  well  of  our  faculties,  and  to 
indulge  sanguine  expectations  concerning  the  improvement  of 
other  parts  of  knowledge.    To  this  I  may  add,  that,  as  mathe- 
matical and  physical  truths  are  perfectly  uninteresting  in  their 
consequences,  the  understanding  readily  yields  its  assent  to  the 
evidence  which  is  presented  to  it;  and  in  this  way  may  be 
expected  to  acquire  the  habit  of  trusting  to  its  own  conclu- 
sions, which  will  contribute  to  fortify  it  against  the  weaknesses 
of  scepticism,  in  the  more  interesting  inquiries  after  moral 
truth  in  which  it  may  afterwards  engage. — STEWART,  DTJGALD. 

Philosophy    of   the    Human    Mind,  Part  3, 
chap.  1,  sect.  3. 

463.  Those  that  can  readily  master  the  difficulties  of  Mathe- 
matics  find  a  considerable  charm  in  the  study,   sometimes 
amounting  to  fascination.    This  is  far  from  universal;  but  the 
subject  contains  elements  of  strong  interest  of  a  kind  that 
constitutes  the  pleasures  of  knowledge.     The  marvellous  de- 
vices for  solving  problems  elate  the  mind  with  the  feeling  of 
intellectual  power;  and  the  innumerable  constructions  of  the 
science  leave  us  lost  in  wonder. — BAIN,  ALEXANDER. 

Education  as  a  Science  (New  York,  1898), 
p.  153. 

451.  Thinking  is  merely  the  comparing  of  ideas,  discerning 
relations  of  likeness  and  of  difference  between  ideas,  and  draw- 
ing inferences.  It  is  seizing  general  truths  on  the  basis  of  clearly 
apprehended  particulars.  It  is  but  generalizing  and  particular- 
izing. Who  will  deny  that  a  child  can  deal  profitably  with  se- 
quences of  ideas  like :  How  many  marbles  are  2  marbles  and  3 
marbles?  2  pencils  and  3  pencils?  2  balls  and  3  balls?  2  chil- 
dren and  3  children?  2  inches  and  3  inches?  2  feet  and  3  feet? 
2  and  3?  Who  has  not  seen  the  countenance  of  some  little 
learner  light  up  at  the  end  of  such  a  series  of  questions  with  the 
exclamation,  "Why  it's  always  that  way.  Isn't  it?"  This  is 
the  glow  of  pleasure  that  the  generalizing  step  always  affords 


THE   VALUE    OF   MATHEMATICS  69 

him  who  takes  the  step  himself.  This  is  the  genuine  life-giving 
joy  which  comes  from  feeling  that  one  can  successfully  take  this 
step.  The  reality  of  such  a  discovery  is  as  great,  and  the  last- 
ing effect  upon  the  mind  of  him  that  makes  it  is  as  sure  as  was 
that  by  which  the  great  Newton  hit  upon  the  generalization  of 
the  law  of  gravitation.  It  is  through  these  thrills  of  discovery 
that  love  to  learn  and  intellectual  pleasure  are  begotten  and  fos- 
tered. Good  arithmetic  teaching  abounds  in  such  opportunities. 

MYERS,  GEORGE. 

Arithmetic   in   Public   Education    (Chicago), 

p.  13. 

455.  A  general  course  in  mathematics  should  be  required  of  all 
officers  for  its  practical  value,  but  no  less  for  its  educational 
value  in  training  the  mind  to  logical  forms  of  thought,  in  devel- 
oping the  sense  of  absolute  truthfulness,  together  with  a  confi- 
dence in  the  accomplishment  of  definite  results  by  definite 

means. — ECHOLS,  C.  P. 

Mathematics  at  West  Point  and  Annapolis; 
U.  S.  Bureau  of  Education,  Bulletin  1912, 
No.  2,  p.  11. 

456.  Exercise  in  the  most  rigorous  thinking  that  is  possible 
will  of  its  own  accord  strengthen  the  sense  of  truth  and  right, 
for  each  advance  in  the  ability  to  distinguish  between  correct 
and  false  thoughts,  each  habit  making  for  rigour  in  thought  de- 
velopment will  increase  in  the  sound  pupil  the  ability  and  the 
wish  to  ascertain  what  is  right  in  life  and  to  defend  it. 

REIDT,  F. 

Anleitung  zum  mathematischen  Unterricht  in 
den  hoheren  Schulen  (Berlin,  1906),  p.  28. 

457.  I  do  not  maintain  that  the  chief  value  of  the  study  of 
arithmetic  consists  in  the  lessons  of  morality  that  arise  from 
this  study.    I  claim  only  that,  to  be  impressed  from  day  to  day, 
that  there  is  something  that  is  right  as  an  answer  to  the  questions 
with  which  one  is  able  to  grapple,  and  that  there  is  a  wrong  an- 
swer— that  there  are  ways  in  which  the  right  answer  can  be 
established  as  right,  that  these  ways  automatically  reject  error 
and  slovenliness,  and  that  the  learner  is  able  himself  to  manipu- 


70  MEMORABILIA  MATHEMATICA 

late  these  ways  and  to  arrive  at  the  establishment  of  the  true 
as  opposed  to  the  untrue,  this  relentless  hewing  to  the  line  and 
stopping  at  the  line,  must  color  distinctly  the  thought  life  of  the 
pupil  with  more  than  a  tinge  of  morality.  ...  To  be  neigh- 
borly with  truth,  to  feel  one's  self  somewhat  facile  in  ways  of 
recognizing  and  establishing  what  is  right,  what  is  correct,  to 
find  the  wrong  persistently  and  unfailingly  rejected  as  of  no 
value,  to  feel  that  one  can  apply  these  ways  for  himself,  that  one 
can  think  and  work  independently,  have  a  real,  a  positive,  and  a 
purifying  effect  upon  moral  character.  They  are  the  quiet, 
steady  undertones  of  the  work  that  always  appeal  to  the  learner 
for  the  sanction  of  his  best  judgment,  and  these  are  the  really 
significant  matters  in  school  work.  It  is  not  the  noise  and  blus- 
ter, not  even  the  dramatics  or  the  polemics  from  the  teacher's 
desk,  that  abide  longest  and  leave  the  deepest  and  stablest  im- 
print upon  character.  It  is  these  still,  small  voices  that  speak 
unmistakably  for  the  right  and  against  the  wrong  and  the  erro- 
neous that  really  form  human  character.  When  the  school 
subjects  are  arranged  on  the  basis  of  the  degree  to  which  they 
contribute  to  the  moral  upbuilding  of  human  character  good 
arithmetic  will  be  well  up  the  list. — MYERS,  GEORGE. 

Arithmetic   in   Public   Education    (Chicago), 
p.  18. 

458.  In  destroying  the  predisposition  to  anger,  science  of  all 
kind  is  useful;  but  the  mathematics  possess  this  property  in  the 
most  eminent  degree. — DR.  RUSH. 

Quoted  in  Day's  Collacon  (London,  no  date). 

459.  The  mathematics  are  the  friends  to  religion,  inasmuch  as 
they  charm  the  passions,  restrain  the  impetuosity  of  the  imagi- 
nation, and  purge  the  mind  from  error  and  prejudice.    Vice  is 
error,  confusion  and  false  reasoning;  and  all  truth  is  more  or  less 
opposite  to  it.    Besides,  mathematical  truth  may  serve  for  a 
pleasant  entertainment  for  those  hours  which  young  men  are  apt 
to  throw  away  upon  their  vices;  the  delightfulness  of  them  being 
such  as  to  make  solitude  not  only  easy  but  desirable. 

ARBUTHNOT,  JOHN. 
Usefulness  of  Mathematical  Learning. 


THE   VALUE   OF  MATHEMATICS  71 

460.  There  is  no  prophet  which  preaches  the  superpersonal 
God  more  plainly  than  mathematics. — CABUS,  PAUL. 

Reflections  on  Magic  Squares;  Monist  (1906), 
p.  147. 

461.  Mathematics  must  subdue  the  flights  of  our  reason;  they 
are  the  staff  of  the  blind;  no  one  can  take  a  step  without  them; 
and  to  them  and  experience  is  due  all  that  is  certain  in  physics. 

VOLTAIRE. 
Oeuvres  Completes  (Paris,  1880),  t.  85,  p.  219. 


CHAPTER   V 

THE   TEACHING   OF   MATHEMATICS 

601.  In  mathematics  two  ends  are  constantly  kept  in  view: 
First,  stimulation  of  the  inventive  faculty,  exercise  of  judgment, 
development  of  logical  reasoning,  and  the  habit  of  concise  state- 
ment; second,  the  association  of  the  branches  of  pure  mathemat- 
ics with  each  other  and  with  applied  science,  that  the  pupil  may 

see  clearly  the  true  relations  of  principles  and  things. 

International  Commission  on  the  Teaching  of 
Mathematics,  American  Report;  U.  S.  Bureau 
of  Education,  Bulktin  1912,  No.  4,  p.  7. 

602.  The  ends  to  be  attained  [in  the  teaching  of  mathematics 
in  the  secondary  schools]  are  the  knowledge  of  a  body  of  geomet- 
rical truths,  the  power  to  draw  correct  inferences  from  given 
premises,  the  power  to  use  algebraic  processes  as  a  means  of  find- 
ing results  in  practical  problems,  and  the  awakening  of  interest 

in  the  science  of  mathematics. 

International  Commission  on  the  Teaching  of 
Mathematics,  American  Report;  U.  S.  Bureau 
of  Education,  Bulktin  1912,  No.  4,  p.  7. 

503.  General  preparatory  instruction  must  continue  to  be  the 
aim  in  the  instruction  at  the  higher  institutions  of  learning.  Ex- 
clusive selection  and  treatment  of  subject  matter  with  reference 

to  specific  avocations  is  disadvantageous. 

Resolution  adopted  by  the  German  Association 
for  the  Advancement  of  Scientific  and  Mathe- 
matical Instruction;  Jahresbericht  der  Deut- 
schen  Mathematiker  Vereinigung  (1896),  p.  41- 

604.  In  the  secondary  schools  mathematics  should  be  a  part 
of  general  culture  and  not  contributory  to  technical  training  of 
any  kind;  it  should  cultivate  space  intuition,  logical  thinking, 
the  power  to  rephrase  hi  clear  language  thoughts  recognized  as 
correct,  and  ethical  and  esthetic  effects;  so  treated,  mathematics 
is  a  quite  indispensable  factor  of  general  education  in  so  far  as 

72 


THE   TEACHING   OF   MATHEMATICS  73 

the  latter  shows  its  traces  in  the  comprehension  of  the  develop- 
ment of  civilization  and  the  ability  to  participate  in  the  further 

tasks  of  civilization. 

Unterrichtsbldtter  fur  Mathematik  und  Natur- 
wissenschaft  (1904),  P-  1%8. 

505.  Indeed,  the  aim  of  teaching  [mathematics]  should  be 
rather  to  strengthen  his  [the  pupil's]  faculties,  and  to  supply  a 
method  of  reasoning  applicable  to  other  subjects,  than  to  fur- 
nish him  with  an  instrument  for  solving  practical  problems. 

MAGNUS,  PHILIP. 

Perry's  Teaching  of  Mathematics  (London, 
1902),  p.  84. 

506.  The  participation  in  the  general  development  of  the  mental 
powers  without  special  reference  to  his  future  vocation  must  be 
recognized  as  the  essential  aim  of  mathematical  instruction. 

REIDT,  F. 

Anleitung  zum  Mathematischen  Unterricht  an 
hdheren  Schulen  (Berlin,  1906),  p.  12. 

507.  I  am  of  the  decided  opinion,  that  mathematical  instruc- 
tion must  have  for  its  first  aim  a  deep  penetration  and  complete 
command  of  abstract  mathematical  theory  together  with  a  clear 
insight  into  the  structure  of  the  system,  and  doubt  not  that  the 
instruction  which  accomplishes  this  is  valuable  and  interesting 
even  if  it  neglects  practical  applications.     If  the  instruction 
sharpens  the  understanding,  if  it  arouses  the  scientific  interest, 
whether  mathematical  or  philosophical,  if  finally  it  calls  into 
life  an  esthetic  feeling  for  the  beauty  of  a  scientific  edifice,  the 
instruction  will  take  on  an  ethical  value  as  well,  provided  that 
with  the  interest  it  awakens  also  the  impulse  toward  scientific 
activity.    I  contend,  therefore,  that  even  without  reference  to 
its  applications  mathematics  in  the  high  schools  has  a  value 
equal  to  that  of  the  other  subjects  of  instruction. 

GOETTING,  E. 

Ueber  das  Lehrziel  im  mathematischen  Unter- 
richt der  hdheren  Realanstalten;  Jahresbericht 
der  Deutschen  Mathematiker  Vereinigung, 
Bd.  2,  p.  192. 


74  MEMORABILIA   MATHEMATICA 

608.  Mathematics  will  not  be  properly  esteemed  in  wider  cir- 
cles until  more  than  the  a  b  c  of  it  is  taught  in  the  schools,  and 
until  the  unfortunate  impression  is  gotten  rid  of  that  mathe- 
matics serves  no  other  purpose  in  instruction  than  the  formal 
training  of  the  mind.     The  aim  of  mathematics  is  its  content,  its 
form  is  a  secondary  consideration  and  need  not  necessarily  be 
that  historic  form  which  is  due  to  the  circumstance  that  mathe- 
matics took  permanent  shape   under  the  influence  of  Greek 
logic. — HANKEL,  H. 

Die   Entwickelung   der   Mathematik   in   den 
letzten  Jahrhunderten  (Tubingen,  1884),  P-  6. 

609.  The  idea  that  aptitude  for  mathematics  is  rarer  than 
aptitude  for  other  subjects  is  merely  an  illusion  which  is  caused 
by  belated  or  neglected  beginners. — HERBART,  J.  F. 

Umriss    padagogischer    Vorlesungen;    Werke 
[Kehrbach]  (Langensaha,  190%),  Bd.  10,  p.  101. 

610.  I  believe  that  the  useful  methods  of  mathematics  are 
easily  to  be  learned  by  quite  young  persons,  just  as  languages 
are  easily  learned  hi  youth.    What  a  wondrous  philosophy  and 
history  underlie  the  use  of  almost  every  word  in  every  lan- 
guage— yet  the  child  learns  to  use  the  word  unconsciously.    No 
doubt  when  such  a  word  was  first  invented  it  was  studied  over 
and  lectured  upon,  just  as  one  might  lecture  now  upon  the  idea 
of  a  rate,  or  the  use  of  Cartesian  co-ordinates,  and  we  may  de- 
pend upon  it  that  children  of  the  future  will  use  the  idea  of 
the  calculus,  and  use  squared  paper  as  readily  as  they  now 
cipher.  .  .  .  When  Egyptian  and  Chaldean  philosophers  spent 
years  in  difficult  calculations,  which  would  now  be  thought  easy 
by  young  children,  doubtless  they  had  the  same  notions  of  the 
depth  of  their  knowledge  that  Sir  William  Thomson  might  now 
have  of  his.    How  is  it,  then,  that  Thomson  gained  his  immense 
knowledge  in  the  time  taken  by  a  Chaldean  philosopher  to  ac- 
quire a  simple  knowledge  of  arithmetic?    The  reason  is  plain. 
Thomson,  when  a  child,  was  taught  in  a  few  years  more  than  all 
that  was  known  three  thousand  years  ago  of  the  properties  of 
numbers.    When  it  is  found  essential  to  a  boy's  future  that  ma- 
chinery should  be  given  to  his  brain,  it  is  given  to  him;  he  is 
taught  to  use  it,  and  his  bright  memory  makes  the  use  of  it  a 


THE  TEACHING  OF  MATHEMATICS  75 

second  nature  to  him;  but  it  is  not  till  after-life  that  he  makes  a 
close  investigation  of  what  there  actually  is  in  his  brain  which 
has  enabled  him  to  do  so  much.  It  is  taken  because  the  child 
has  much  faith.  In  after  years  he  will  accept  nothing  without 
careful  consideration.  The  machinery  given  to  the  brain  of 
children  is  getting  more  and  more  complicated  as  time  goes  on; 
but  there  is  really  no  reason  why  it  should  not  be  taken  in  as 
early,  and  used  as  readily,  as  were  the  axioms  of  childish  educa- 
tion in  ancient  Chaldea. — PERRY,  JOHN. 

The  Teaching  of  Mathematics  (London,  1902}, 

p.  14. 

517.  The  ancients  devoted  a  lifetime  to  the  study  of  arith- 
metic; it  required  days  to  extract  a  square  root  or  to  multiply 
two  numbers  together.  Is  there  any  harm  in  skipping  all  that, 
in  letting  the  school  boy  learn  multiplication  sums,  and  in  start- 
ing his  more  abstract  reasoning  at  a  more  advanced  point? 
Where  would  be  the  harm  in  letting  the  boy  assume  the  truth  of 
many  propositions  of  the  first  four  books  of  Euclid,  letting  him 
assume  their  truth  partly  by  faith,  partly  by  trial?  Giving  him 
the  whole  fifth  book  of  Euclid  by  simple  algebra?  Letting  him 
assume  the  sixth  as  axiomatic?  Letting  him,  in  fact,  begin  his 
severer  studies  where  he  is  now  in  the  habit  of  leaving  off?  We 
do  much  less  orthodox  things.  Every  here  and  there  in  one's 
mathematical  studies  one  makes  exceedingly  large  assumptions, 
because  the  methodical  study  would  be  ridiculous  even  in  the 
eyes  of  the  most  pedantic  of  teachers.  I  can  imagine  a  whole  year 
devoted  to  the  philosophical  study  of  many  things  that  a  student 
now  takes  in  his  stride  without  trouble.  The  present  method  of 
training  the  mind  of  a  mathematical  teacher  causes  it  to  strain 
at  gnats  and  to  swallow  camels.  Such  gnats  are  most  of  the 
propositions  of  the  sixth  book  of  Euclid;  propositions  generally 
about  incommensurables;  the  use  of  arithmetic  in  geometry;  the 
parallelogram  of  forces,  etc.,  decimals. — PERRY,  JOHN. 

The  Teaching  of  Mathematics  (London,  1904), 

p.  12. 

612.  The  teaching  of  elementary  mathematics  should  be  con- 
ducted so  that  the  way  should  be  prepared  for  the  building  upon 
them  of  the  higher  mathematics.  The  teacher  should  always 


76  MEMORABILIA   MATHEMATICA 

bear  in  mind  and  look  forward  to  what  is  to  come  after.  The 
pupil  should  not  be  taught  what  may  be  sufficient  for  the  tune, 
but  will  lead  to  difficulties  in  the  future.  ...  I  think  the  fault 
in  teaching  arithmetic  is  that  of  not  attending  to  general 
principles  and  teaching  instead  of  particular  rules.  ...  I  am 
inclined  to  attack  the  teaching  of  mathematics  on  the  grounds 
that  it  does  not  dwell  sufficiently  on  a  few  general  axiomatic 
principles. — HUDSON,  W.  H.  H. 

Perry's   Teaching  of  Mathematics    (London, 

1904),  P-  33. 

513.  "Mathematics  in  Prussia!    Ah,  sir,  they  teach  mathe- 
matics in  Prussia  as  you  teach  your  boys  rowing  in  England: 
they  are  trained  by  men  who  have  been  trained  by  men  who 
have  themselves  been  trained  for  generations  back." 

LANGLEY,  E.  M. 

Perry's   Teaching   of  Mathematics    (London, 
1904),  P-  43. 

514.  A  superficial  knowledge  of  mathematics  may  lead  to  the 
belief  that  this  subject  can  be  taught  incidentally,  and  that  ex- 
ercises akin  to  counting  the  petals  of  flowers  or  the  legs  of  a 
grasshopper  are  mathematical.    Such  work  ignores  the  funda- 
mental idea  out  of  which  quantitative  reasoning  grows — the 
equality  of  magnitudes.     It  leaves  the  pupil  unaware  of  that 
relativity  which  is  the  essence  of  mathematical  science.    Nu- 
merical statements  are  frequently  required  in  the  study  of  nat- 
ural history,  but  to  jepeat  these  as  a  drill  upon  numbers  will 
scarcely  lend  charm  to  these  studies,  and  certainly  will  not 
result  in  mathematical  knowledge. — SPEER,  W.  W. 

Primary  Arithmetic  (Boston,  1897),  pp.  26-27. 

515.  Mathematics  is  no  more  the  art  of  reckoning  and  com- 
putation than  architecture  is  the  art  of  making  bricks  or  hewing 
wood,  no  more  than  painting  is  the  art  of  mixing  colors  on  a 
palette,  no  more  than  the  science  of  geology  is  the  art  of  breaking 
rocks,  or  the  science  of  anatomy  the  art  of  butchering. 

KEYSER,  C.  J. 

Lectures  on  Science,  Philosophy  and  Art  (New 
York,  1908),  p.  29. 


THE   TEACHING   OF   MATHEMATICS  77 

516.  The  study  of  mathematics — from  ordinary  reckoning  up 
to  the  higher  processes — must  be  connected  with  knowledge  of 
nature,  and  at  the  same  time  with  experience,  that  it  may  enter 

the  pupil's  circle  of  thought. — HERBART,  J.  F. 

Letters  and  Lectures  on  Education  [Felkin] 
(London,  1908),  p.  117. 

517.  First,  as  concerns  the  success  of  teaching  mathematics. 
No  instruction  in  the  high  schools  is  as  difficult  as  that  of  mathe- 
matics, since  the  large  majority  of  students  are  at  first  decidedly 
disinclined  to  be  harnessed  into  the  rigid  framework  of  logical 
conclusions.    The  interest  of  young  people  is  won  much  more 
easily,  if  sense-objects  are  made  the  starting  point  and  the  tran- 
sition to  abstract  formulation  is  brought  about  gradually.    For 
this  reason  it  is  psychologically  quite  correct  to  follow  this  course. 

Not  less  to  be  recommended  is  this  course  if  we  inquire  into 
the  essential  purpose  of  mathematical  instruction.  Formerly  it 
was  too  exclusively  held  that  this  purpose  is  to  sharpen  the 
understanding.  Surely  another  important  end  is  to  implant  in 
the  student  the  conviction  that  correct  thinking  based  on  true 
premises  secures  mastery  over  the  outer  world.  To  accomplish  this 
the  outer  world  must  receive  its  share  of  attention  from  the  very- 
beginning. 

Doubtless  this  is  true  but  there  is  a  danger  which  needs  point- 
ing out.  It  is  as  in  the  case  of  language  teaching  where  the 
modern  tendency  is  to  secure  in  addition  to  grammar  also  an 
understanding  of  the  authors.  The  danger  lies  in  grammar  being 
completely  set  aside  leaving  the  subject  without  its  indispensa- 
ble solid  basis.  Just  so  in  the  teaching  of  mathematics  it  is 
possible  to  accumulate  interesting  applications  to  such  an  extent 
as  to  stunt  the  essential  logical  development.  This  should  in  no 
wise  be  permitted,  for  thus  the  kernel  of  the  whole  matter  is  lost. 
Therefore :  We  do  want  throughout  a  quickening  of  mathemat- 
ical instruction  by  the  introduction  of  applications,  but  we  do 
not  want  that  the  pendulum,  which  in  former  decades  may  have 
inclined  too  much  toward  the  abstract  side,  should  now  swing 
to  the  other  extreme;  we  would  rather  pursue  the  proper  middle 
course. — KLEIN,  FELIX. 

Ueber  den  Mathematischen  Unterricht  an  den 
hoheren  Schulen;  Jahresbericht  der  Deutschen 
Mathematiker  Vereinigung,  Bd.  11,  p.  181. 


78  MEMORABILIA   MATHEMATICA 

518.  It  is  above  all  the  duty  of  the  methodical  text-book  to 
adapt  itself  to  the  pupil's  power  of  comprehension,  only  chal- 
lenging his  higher  efforts  with  the  increasing  development  of  his 
imagination,  his  logical  power  and  the  ability  of  abstraction. 
This  indeed  constitutes  a  test  of  the  art  of  teaching,  it  is  here 
where  pedagogic  tact  becomes  manifest.    In  reference  to  the 
axioms,  caution  is  necessary.    It  should  be  pointed  out  compara- 
tively early,  hi  how  far  the  mathematical  body  differs  from  the 
material  body.     Furthermore,  since  mathematical  bodies  are 
really  portions  of  space,  this  space  is  to  be  conceived  as  mathe- 
matical space  and  to  be  clearly  distinguished  from  real  or  phys- 
ical space.    Gradually  the  student  will  become  conscious  that 
the  portion  of  the  real  space  which  lies  beyond  the  visible  stellar 
universe  is  not  cognizable  through  the  senses,  that  we  know 
nothing  of  its  properties  and  consequently  have  no  basis  for 
judgments  concerning  it.     Mathematical  space,  on  the  other 
hand,  may  be  subjected  to  conditions,  for  instance,  we  may  con- 
dition its  properties  at  infinity,  and  these  conditions  constitute 
the  axioms,  say  the  Euclidean  axioms.    But  every  student  will 
require  years  before  the  conviction  of  the  truth  of  this  last  state- 
ment will  force  itself  upon  him. — HOLZMULLER,  GUSTAV. 

Methodisches  Lehrbuch  der  Elementar-Mathe- 
matik  (Leipzig,  1904),  Teil  I,  Vorwort,  pp.  4~5- 

519.  Like  almost  every  subject  of  human  interest,  this  one 
[mathematics]  is  just  as  easy  or  as  difficult  as  we  choose  to  make 
it.    A  lifetime  may  be  spent  by  a  philosopher  in  discussing  the 
truth  of  the  simplest  axiom.    The  simplest  fact  as  to  our  exist- 
ence may  fill  us  with  such  wonder  that  our  minds  will  remain 
overwhelmed  with  wonder  all  the  time.    A  Scotch  ploughman 
makes  a  working  religion  out  of  a  system  which  appalls  a  mental 
philosopher.    Some  boys  of  ten  years  of  age  study  the  methods 
of  the  differential  calculus;  other  much  cleverer  boys  working  at 
mathematics  to  the  age  of  nineteen  have  a  difficulty  in  compre- 
hending the  fundamental  ideas  of  the  calculus. — PERRY,  JOHN. 

The  Teaching  of  Mathematics  (London,  1902), 
pp.  19-20. 

520.  Poor  teaching  leads  to  the  inevitable  idea  that  the  sub- 
ject [mathematics]  is  only  adapted  to  peculiar  minds,  when  it  is 


THE   TEACHING    OF   MATHEMATICS  79 

the  one  universal  science  and  the  one  whose  four  ground-rules 
are  taught  us  almost  in  infancy  and  reappear  in  the  motions  of 
the  universe. — SAFFOBD,  T.  H. 

Mathematical  Teaching  (Boston,  1907),  p.  19. 

521.  The  number  of  mathematical  students  .  .  .  would  be 
much  augmented  if  those  who  hold  the  highest  rank  in  science 
would  condescend  to  give  more  effective  assistance  in  clearing 
the  elements  of  the  difficulties  which  they  present. 

DE  MORGAN,  A. 

Study  and  Difficulties  of  Mathematics  (Chicago, 
1902),  Preface. 

522.  He  that  could  teach  mathematics  well,  would  not  be  a 
bad  teacher  in  any  of  the  rest  [physics,  chemistry,  biology,  psy- 
chology] unless  by  the  accident  of  total  inaptitude  for  experi- 
mental illustration;  while  the  mere  experimentalist  is  likely  to 
fall  into  the  error  of  missing  the  essential  condition  of  science  as 
reasoned  truth;  not  to  speak  of  the  danger  of  making  the  in- 
struction an  affair  of  sensation,  glitter,  or  pyrotechnic  show. 

BAIN,  ALEXANDER. 

Education  as  a  Science  (New   York,  1898), 
p.  298. 

523.  I  should  like  to  draw  attention  to  the  inexhaustible 
variety  of  the  problems  and  exercises  which  it  [mathematics] 
furnishes;  these  may  be  graduated  to  precisely  the  amount  of 
attainment  which  may  be  possessed,  while  yet  retaining  an  in- 
terest and  value.    It  seems  to  me  that  no  other  branch  of  study 
at  all  compares  with  mathematics  in  this.    When  we  propose  a 
deduction  to  a  beginner  we  give  him  an  exercise  in  many  cases 
that  would  have  been  admired  in  the  vigorous  days  of  Greek 
geometry.    Although  grammatical  exercises  are  well  suited  to 
insure  the  great  benefits  connected  with  the  study  of  languages, 
yet  these  exercises  seem  to  me  stiff  and  artificial  in  comparison 
with  the  problems  of  mathematics.    It  is  not  absurd  to  maintain 
that  Euclid  and  Apollonius  would  have  regarded  with  interest 
many  of  the  elegant  deductions  which  are  invented  for  the  use 
of  our  students  in  geometry;  but  it  seems  scarcely  conceivable 


80  MEMORABILIA   MATHEMATICA 

that  the  great  masters  in  any  other  line  of  study  could  conde- 
scend to  give  a  moment's  attention  to  the  elementary  books  of 
the  beginner. — TODHUNTER,  ISAAC. 

Conflict  of  Studies  (London,  1873),  pp.  10-11. 

624.  The  visible  figures  by  which  principles  are  illustrated 
should,  so  far  as  possible,  have  no  accessories.    They  should  be 
magnitudes  pure  and  simple,  so  that  the  thought  of  the  pupil 
may  not  be  distracted,  and  that  he  may  know  what  features  of 
the  thing  represented  he  is  to  pay  attention  to. 

Report  of  the  Committee  of  Ten  on  Secondary 
School  Subjects,  (New  York,  1894),  P-  109. 

625.  Geometrical  reasoning,  and  arithmetical  process,  have 
each  its  own  office:  to  mix  the  two  in  elementary  instruction, 
is  injurious  to  the  proper  acquisition  of  both. 

DE  MORGAN,  A. 

Trigonometry  and  Double  Algebra    (London, 
1849),  p.  92. 

526.  Equations  are  Expressions  of  Arithmetical  Computa- 
tion, and  properly  have  no  place  in  Geometry,  except  as  far  as 
Quantities  truly  Geometrical  (that  is,  Lines,  Surfaces,  Solids, 
and  Proportions)  may  be  said  to  be  some  equal  to  others.  Mul- 
tiplications, Divisions,  and  such  sort  of  Computations,  are 
newly  received  into  Geometry,  and  that  unwarily,  and  contrary 
to  the  first  Design  of  this  Science.  For  whosoever  considers  the 
Construction  of  a  Problem  by  a  right  Line  and  a  Circle,  found 
out  by  the  first  Geometricians,  will  easily  perceive  that  Geome- 
try was  invented  that  we  might  expeditiously  avoid,  by  drawing 
Lines,  the  Tediousness  of  Computation.  Therefore  these  two 
Sciences  ought  not  to  be  confounded.  The  Ancients  did  so  in- 
dustriously distinguish  them  from  one  another,  that  they  never 
introduced  Arithmetical  Terms  into  Geometry.  And  the  Mod- 
erns, by  confounding  both,  have  lost  the  Simplicity  in  which  all 
the  Elegance  of  Geometry  consists.  Wherefore  that  is  Arith- 
metically more  simple  which  is  determined  by  the  more  simple 
Equation,  but  that  is  Geometrically  more  simple  which  is  deter- 
mined by  the  more  simple  drawing  of  Lines;  and  in  Geometry, 


THE    TEACHING   OF   MATHEMATICS  81 

that  ought  to  be  reckoned  best  which  is  geometrically  most 

simple. — NEWTON. 

On  the  Linear  Construction  of  Equations; 
Universal  Arithmetic  (London,  1769),  Vol.  2, 
p.  470. 

627.  As  long  as  algebra  and  geometry  proceeded  along  sepa- 
rate paths,  their  advance  was  slow  and  their  applications 
limited. 

But  when  these  sciences  joined  company,  they  drew  from 
each  other  fresh  vitality  and  thenceforward  marched  on  at  a 
rapid  pace  toward  perfection. — LAGRANGE. 

LeQons  filementaires  sur  les  Mathematiques, 
Leqon  cinquieme.  [McCormack]. 

528.  The  greatest  enemy  to  true  arithmetic  work  is  found  in 
so-called  practical  or  illustrative  problems,  which  are  freely 
given  to  our  pupils,  of  a  degree  of  difficulty  and  complexity  alto- 
gether unsuited  to  their  age  and  mental  development.    ...  I 
am,  myself,  no  bad  mathematician,  and  all  the  reasoning  powers 
with  which  nature  endowed  me  have  long  been  as  fully  devel- 
oped as  they  are  ever  likely  to  be;  but  I  have,  not  infrequently, 
been  puzzled,  and  at  tunes  foiled,  by  the  subtle  logical  difficulty 
running  through  one  of  these  problems,  given  to  my  own  chil- 
dren.   The  head-master  of  one  of  our  Boston  high  schools  con- 
fessed to  me  that  he  had  sometimes  been  unable  to  unravel 
one  of  these  tangled  skeins,  in  trying  to  help  his  own  daughter 
through  her  evening's  work.     During  this  summer,  Dr.  Fair- 
bairn,  the  distinguished  head  of  one  of  the  colleges  of  Oxford, 
England,  told  me  that  not  only  had  he  himself  encountered  a 
similar  difficulty,  in  the  case  of  his  own  children,  but  that,  on 
one  occasion,  having  as  his  guest  one  of  the  first  mathematicians 
of  England,  the  two  together  had  been  completely  puzzled  by 
one  of  these  arithmetical  conundrums. — WALKER,  F.  A. 

Discussions  in  Education  (New  York,  1899), 
pp.  253-254. 

529.  It  is  often  assumed  that  because  the  young  child  is  not 
competent  to  study  geometry  systematically  he  need  be  taught 
nothing  geometrical;  that  because  it  would  be  foolish  to  present 


82  MEMORABILIA   MATHEMATICA 

to  him  physics  and  mechanics  as  sciences  it  is  useless  to  present 
to  him  any  physical  or  mechanical  principles. 

An  error  of  like  origin,  which  has  wrought  incalculable  mis- 
chief, denies  to  the  scholar  the  use  of  the  symbols  and  methods 
of  algebra  in  connection  with  his  early  essays  in  numbers  be- 
cause, forsooth,  he  is  not  as  yet  capable  of  mastering  quad- 
ratics! .  .  .  The  whole  infant  generation,  wrestling  with  arith- 
metic, seek  for  a  sign  and  groan  and  travail  together  in  pain  for 
the  want  of  it;  but  no  sign  is  given  them  save  the  sign  of  the 
prophet  Jonah,  the  withered  gourd,  fruitless  endeavor,  wasted 
strength. — WALKER,  F.  A. 

Industrial  Education;  Discussions  in  Educa- 
tion (New  York,  1899),  p.  132. 

530.  Particular  and  contingent  inventions  in  the  solution  of 
problems,  which,  though  many  tunes  more  concise  than  a  gen- 
eral method  would  allow,  yet,  in  my  judgment,  are  less  proper  to 
instruct  a  learner,  as  acrostics,  and  such  kind  of  artificial  poetry, 
though  never  so  excellent,  would  be  but  improper  examples  to 
instruct  one  that  aims  at  Ovidean  poetry. — NEWTON,  ISAAC. 

Letter  to  Collins,  1670;  Maccles -field,  Corre- 
spondence of  Scientific  Men  (Oxford,  1841), 
Vol.  2,  p.  807. 

531.  The  logic  of  the  subject  [algebra],  which,  both  educa- 
tionally and  scientifically  speaking,  is  the  most  important  part 
of  it,  is  wholly  neglected.    The  whole  training  consists  in  exam- 
ple grinding.    What  should  have  been  merely  the  help  to  attain 
the  end  has  become  the  end  itself.    The  result  is  that  algebra,  as 
we  teach  it,  is  neither  an  art  nor  a  science,  but  an  ill-digested 
farrago  of  rules,  whose  object  is  the  solution  of  examination 
problems.  .  .  .  The  result,  so  far  as  problems  worked  in  exam- 
inations go,  is,  after  all,  very  miserable,  as  the  reiterated  com- 
plaints of  examiners  show;  the  effect  on  the  examinee  is  a  well- 
known  enervation  of  mind,  an  almost  incurable  superficiality, 
which  might  be  called  Problematic  Paralysis — a  disease  which 
unfits  a  man  to  follow  an  argument  extending  beyond  the 
length  of  a  printed  octavo  page. — CHRYSTAL,  GEORGE. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science,  1885;  Nature, 
Vol.  32,  pp.  447-448. 


THE   TEACHING   OF   MATHEMATICS  83 

532.  It  is  a  serious  question  whether  America,  following  Eng- 
land's lead,  has  not  gone  into  problem-solving  too  extensively. 
Certain  it  is  that  we  are  producing  no  text-books  in  which  the 
theory  is  presented  in  the  delightful  style  which  characterizes 
many  of  the  French  works  .  .  .  ,  or  those  of  the  recent  Italian 
school,  or,  indeed,  those  of  the  continental  writers  in  general. 

SMITH,  D.  E. 

The  Teaching  of  Elementary  Mathematics 
(New  York,  1902),  p.  219. 

533.  The  problem  for  a  writer  of  a  text-book  has  come  now, 
in  fact,  to  be  this — to  write  a  book  so  neatly  trimmed  and  com- 
pacted that  no  coach,  on  looking  through  it,  can  mark  a  single 
passage  which  the  candidate  for  a  minimum  pass  can  safely 
omit.    Some  of  these  text-books  I  have  seen,  where  the  scientific 
matter  has  been,  like  the  lady's  waist  in  the  nursery  song,  com- 
pressed "so  gent  and  sma',"  that  the  thickness  barely,  if  at  all, 
surpasses  what  is  devoted  to  the  publisher's  advertisements. 
We  shall  return,  I  verily  believe,  to  the  Compendium  of  Mar- 
tianus  Capella.    The  result  of  all  this  is  that  science,  in  the  hands 
of  specialists,  soars  higher  and  higher  into  the  light  of  day, 
while  educators  and  the  educated  are  left  more  and  more  to  wan- 
der in  primeval  darkness. — CHEYSTAL,  GEORGE. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science,  1885;  Nature, 
Vol.  32,  p.  448- 

1  534.  Some  persons  have  contended  that  mathematics  ought 
to  be  taught  by  making  the  illustrations  obvious  to  the  senses. 
Nothing  can  be  more  absurd  or  injurious:  it  ought  to  be  our 
never-ceasing  effort  to  make  people  think,  not  feel. 

COLERIDGE,  S.  T. 
Lectures  on  Shakespere  (Bohn  Library),  p.  52. 

535.  I  have  come  to  the  conclusion  that  the  exertion,  without 
which  a  knowledge  of  mathematics  cannot  be  acquired,  is  not 
materially  increased  by  logical  rigor  in  the  method  of  instruc- 
tion.— PRINGSHEIM,  ALFRED. 

Jahresbericht  der  Deutschen  Mathematiker 
Vereinigung  (1898),  p.  143. 


84  MEMORABILIA   MATHEMATICA 

636.  The  only  way  in  which  to  treat  the  elements  of  an  exact 
and  rigorous  science  is  to  apply  to  them  all  the  rigor  and  exact- 
ness possible. — D'ALEMBERT. 

Quoted  by    De  Morgan:    Trigonometry    and 
Double  Algebra  (London,  1849),  Title  page. 

637.  It  is  an  error  to  believe  that  rigor  in  proof  is  an  enemy 
of  simplicity.    On  the  contrary  we  find  it  confirmed  by  numer- 
ous examples  that  the  rigorous  method  is  at  the  same  time  the 
simpler  and  the  more  easily  comprehended.    The  very  effort  for 
rigor  forces  us  to  find  out  simpler  methods  of  proof. 

HILBERT,  D. 

Mathematical   Problems;   Bulletin   American 
Mathematical   Society,    Vol.   8,    p.   441- 

538.  Few  will  deny  that  even  in  the  first  scientific  instruction 
in  mathematics  the  most  rigorous  method  is  to  be  given  prefer- 
ence over  all  others.  Especially  will  every  teacher  prefer  a  con- 
sistent proof  to  one  which  is  based  on  fallacies  or  proceeds  hi  a 
vicious  circle,  indeed  it  will  be  morally  impossible  for  the  teacher 
to  present  a  proof  of  the  latter  kind  consciously  and  thus  in  a 
sense  deceive  his  pupils.  Notwithstanding  these  objectionable 
so-called  proofs,  so  far  as  the  foundation  and  the  development 
of  the  system  is  concerned,  predominate  in  our  textbooks  to  the 
present  time.  Perhaps  it  will  be  answered,  that  rigorous  proof 
is  found  too  difficult  for  the  pupil's  power  of  comprehension. 
Should  this  be  anywhere  the  case, — which  would  only  indicate 
some  defect  in  the  plan  or  treatment  of  the  whole, — the  only 
remedy  would  be  to  merely  state  the  theorem  in  a  historic  way, 
and  forego  a  proof  with  the  frank  confession  that  no  proof  has 
been  found  which  could  be  comprehended  by  the  pupil;  a  remedy 
which  is  ever  doubtful  and  should  only  be  applied  in  the  case  of 
extreme  necessity.  But  this  remedy  is  to  be  preferred  to  a  proof 
which  is  no  proof,  and  is  therefore  either  wholly  unintelligible  to 
the  pupil,  or  deceives  him  with  an  appearance  of  knowledge 
which  opens  the  door  to  all  superficiality  and  lack  of  scientific 
method. — GRASSMANN,  HERMANN. 

Stucke  aus  dem  Lehrbuche  der  Arithmetik; 
Werke,    Bd.    2    (Leipsig,    1904),    P-   296. 


THE   TEACHING   OF   MATHEMATICS  85 

539.  The  average  English  author  [of  mathematical  texts] 
leaves  one  under  the  impression  that  he  has  made  a  bargain 
with  his  reader  to  put  before  him  the  truth,  the  greater  part  of 
the  truth,  and  nothing  but  the  truth;  and  that  if  he  has  put  the 
facts  of  his  subject  into  his  book,  however  difficult  it  may  be  to 
unearth  them,  he  has  fulfilled  his  contract  with  his  reader.    This 
is  a  very  much  mistaken  view,  because  effective  teaching  requires 
a  great  deal  more  than  a  bare  recitation  of  facts,  even  if  these  are 
duly  set  forth  in  logical  order — as  in  English  books  they  often 
are  not.    The  probable  difficulties  which  will  occur  to  the  stu- 
dent, the  objections  which  the  intelligent  student  will  naturally 
and  necessarily  raise  to  some  statement  of  fact  or  theory — these 
things  our  authors  seldom  or  never  notice,  and  yet  a  recognition 
and  anticipation  of  them  by  the  author  would  be  often  of  price- 
less value  to  the  student.    Again,  a  touch  of  humour  (strange  as 
the  contention  may  seem)  in  mathematical  works  is  not  only 
possible  with  perfect  propriety,  but  very  helpful;  and  I  could 
give  instances  of  this  even  from  the  pure  mathematics  of  Salmon 
and  the  physics  of  Clerk  Maxwell. — MINCHIN,  G.  M. 

Perry's  Teaching  of  Mathematics  (London, 
1902},  pp.  59-61. 

540.  Remember  this,  the  rule  for  giving  an  extempore  lecture 
is — let  the  the  mind  rest  from  the  subject  entirely  for  an  interval 
preceding  the  lecture,  after  the  notes  are  prepared;  the  thoughts 
will  ferment  without  your  knowing  it,  and  enter  into  new  com- 
binations; but  if  you  keep  the  mind  active  upon  the  subject  up 
to  the  moment,  the  subject  will  not  ferment  but  stupefy. 

DE  MORGAN,  A. 

Letter  to  Hamilton;  Graves:  Life  of  W.  R. 
Hamilton  (New  York,  1882-1889),  Vol.  3, 
p.  487. 


CHAPTER  VI 

STUDY   AND   RESEARCH   IN   MATHEMATICS 

601.  The  first  thing  to  be  attended  to  in  reading  any  algebraic 
treatise  is  the  gaining  a  perfect  understanding  of  the  different 
processes  there  exhibited,  and  of  their  connection  with  one  an- 
other.   This  cannot  be  attained  by  the  mere  reading  of  the  book, 
however  great  the  attention  which  may  be  given.    It  is  impos- 
sible in  a  mathematical  work  to  fill  up  every  process  in  the  man- 
ner in  which  it  must  be  filled  up  in  the  mind  of  the  student  before 
he  can  be  said  to  have  completely  mastered  it.    Many  results 
must  be  given  of  which  the  details  are  suppressed,  such  are  the 
additions,  multiplications,  extractions  of  square  roots,  etc.,  with 
which  the  investigations  abound.    These  must  not  be  taken  on 
trust  by  the  student,  but  must  be  worked  out  by  his  own  pen, 
which  must  never  be  out  of  his  own  hand  while  engaged  in  any 
mathematical  process. — DE  MORGAN,  A. 

Study  and  Difficulties  of  Mathematics  (Chi- 
cago, 1902},  chap.  12. 

602.  The  student  should  not  lose  any  opportunity  of  exer- 
cising himself  in  numerical  calculation  and  particularly  in  the 
use  of  logarithmic  tables.    His  power  of  applying  mathematics 
to  questions  of  practical  utility  is  in  direct  proportion  to  the 
facility  which  he  possesses  in  computation. — DE  MORGAN,  A. 

Study  and  Difficulties  of  Mathematics  (Chi- 
cago, 1902),  chap.  12. 

603.  The  examples  which  a  beginner  should  choose  for  prac- 
tice should  be  simple  and  should  not  contain  very  large  num- 
bers.   The  powers  of  the  mind  cannot  be  directed  to  two  things 
at  once;  if  the  complexity  of  the  numbers  used  requires  all  the 
student's  attention,  he  cannot  observe  the  principle  of  the  rule 
which  he  is  following. — DE  MORGAN,  A. 

Study  and  Difficulties  of  Mathematics  (Chi- 
cago, 1902),  chap.  8. 
86 


STUDY   AND    RESEARCH   IN   MATHEMATICS  87 

604.  Euclid  and  Archimedes  are  allowed  to  be  knowing,  and 
to  have  demonstrated  what  they  say:   and  yet  whosoever  shall 
read  over  their  writings  without  perceiving  the  connection  of 
their  proofs,  and  seeing  what  they  show,  though  he  may  under- 
stand all  their  words,  yet  he  is  not  the  more  knowing.    He  may 
believe,  indeed,  but  does  not  know  what  they  say,  and  so  is  not 
advanced  one  jot  in  mathematical  knowledge  by  all  his  reading 
of  those  approved  mathematicians. — LOCKE,  JOHN. 

Conduct  of  the  Understanding,  sect.  24. 

605.  The  student  should  read  his  author  with  the  most  sus- 
tained attention,  in  order  to  discover  the  meaning  of  every  sen- 
tence.   If  the  book  is  well  written,  it  will  endure  and  repay  his 
close  attention:    the  text  ought  to  be  fairly  intelligible,  even 
without  illustrative  examples.     Often,  far  too  often,  a  reader 
hurries  over  the  text  without  any  sincere  and  vigorous  effort  to 
understand  it;  and  rushes  to  some  example  to  clear  up  what 
ought  not  to  have  been  obscure,  if  it  had  been  adequately  con- 
sidered.   The  habit  of  scrupulously  investigating  the  text  seems 
to  me  important  on  several  grounds.    The  close  scrutiny  of  lan- 
guage is  a  very  valuable  exercise  both  for  studious  and  practical 
life.     In  the  higher  departments  of  mathematics  the  habit  is 
indispensable:    in  the  long  investigations  which  occur  there  it 
would  be  impossible  to  interpose  illustrative  examples  at  every 
stage,  the  student  must  therefore  encounter  and  master,  sen- 
tence by  sentence,  an  extensive  and  complicated  argument. 

TODHUNTER,    ISAAC. 

Private   Study   of  Mathematics;   Conflict   of 
Studies  and  other  Essays  (London,  1873),  p.  67. 

606.  It  must  happen  that  in  some  cases  the  author  is  not 
understood,  or  is  very  imperfectly  understood;  and  the  question 
is  what  is  to  be  done.    After  giving  a  reasonable  amount  of  at- 
tention to  the  passage,  let  the  student  pass  on,  reserving  the 
obscurity  for  future  efforts.  .  .  .  The  natural  tendency  of  soli- 
tary students,  I  believe,  is  not  to  hurry  away  prematurely  from 
a  hard  passage,  but  to  hang  far  too  long  over  it;  the  just  pride 
that  does  not  like  to  acknowledge  defeat,  and  the  strong  will 
that  cannot  endure  to  be  thwarted,  both  urge  to  a  continuance 
of  effort  even  when  success  seems  hopeless.    It  is  only  by  experi- 


88  MEMORABILIA   MATHEMATICA 

ence  we  gain  the  conviction  that  when  the  mind  is  thoroughly 
fatigued  it  has  neither  the  power  to  continue  with  advantage  its 
course  in  an  assigned  direction,  nor  elasticity  to  strike  out  a  new 
path;  but  that,  on  the  other  hand,  after  being  withdrawn  for  a 
time  from  the  pursuit,  it  may  return  and  gain  the  desired  end. 

TODHUNTER,    ISAAC. 

Private  Study  of  Mathematics;  Conflict  of 
Studies  and  other  Essays  (London,  1878), 
p.  68. 

607.  Every  mathematical  book  that  is  worth  reading  must  be 
read  "backwards  and  forwards,"  if  I  may  use  the  expression. 
I  would  modify  Lagrange's  advice  a  little  and  say,  "Go  on,  but 
often  return  to  strengthen  your  faith."    When  you  come  on  a 
hard  or  dreary  passage,  pass  it  over;  and  come  back  to  it  after 
you  have  seen  its  importance  or  found  the  need  for  it  further  on. 

CHRYSTAL,  GEORGE. 
Algebra,  Part  2  (Edinburgh,  1889),   Preface, 
p.  8. 

608.  The  large  collection  of  problems  which  our  modern 
Cambridge  books  supply  will  be  found  to  be  almost  an  exclusive 
peculiarity  of  these  books;  such  collections  scarcely  exist  in  for- 
eign treatises  on  mathematics,  nor  even  hi  English  treatises  of 
an  earlier  date.    This  fact  shows,  I  think,  that  a  knowledge  of 
mathematics  may  be  gained  without  the  perpetual  working  of 
examples.  .  .  .  Do  not  trouble  yourselves  with  the  examples, 
make  it  your  main  business,  I  might  almost  say  your  exclusive 
business,  to  understand  the  text  of  your  author. 

TODHUNTER,  ISAAC. 

Private  Study  of  Mathematics;  Conflict  of  Stud- 
ies and  other  Essays  (London,  1878),  p.  74- 

609.  In  my  opinion  the  English  excel  in  the  art  of  writing 
text-books  for  mathematical  teaching;  as  regards  the  clear  expo- 
sition of  theories  and  the  abundance  of  excellent  examples,  care- 
fully selected,  very  few  books  exist  in  other  countries  which  can 
compete  with  those  of  Salmon  and  many  other  distinguished 
English  authors  that  could  be  named. — CREMONA,  L. 

Protective    Geometry    [Leudesdorf]    (Oxford, 
1885),  Preface. 


STUDY   AND   RESEARCH    IN   MATHEMATICS  89 

610.  The  solution  of  fallacies,  which  give  rise  to  absurdities, 
should  be  to  him  who  is  not  a  first  beginner  in  mathematics  an 
excellent  means  of  testing  for  a  proper  intelligible  insight  into 
mathematical  truth,  of  sharpening  the  wit,  and  of  confining  the 
judgment  and  reason  within  strictly  orderly  limits. — VIOLA,  J. 

Mathematische  Sophismen  (Wien,  1864),  Vor- 
wort. 

611.  Success  in  the  solution  of  a  problem  generally  depends 
in  a  great  measure  on  the  selection  of  the  most  appropriate 
method  of  approaching  it;  many  properties  of  conic  sections 
(for  instance)  being  demonstrable  by  a  few  steps  of  pure  geome- 
try which  would  involve  the  most  laborious  operations  with 
trilinear  co-ordinates,  while  other  properties  are  almost  self- 
evident  under  the  method  of  trilinear  co-ordinates,  which  it 
would  perhaps  be  actually  impossible  to  prove  by  the  old 
geometry. — WHITWORTH,  W.  A. 

Modern  Analytic  Geometry  (Cambridge,  1866), 
p.  154. 

612.  The  deep  study  of  nature  is  the  most  fruitful  source  of 
mathematical  discoveries.     By  offering  to  research  a  definite 
end,  this  study  has  the  advantage  of  excluding  vague  questions 
and  useless  calculations;  besides  it  is  a  sure  means  of  forming 
analysis  itself  and  of  discovering  the  elements  which  it  most 
concerns  us  to  know,  and  which  natural  science  ought  always  to 
conserve. — FOURIER,  J. 

Th&orie  Analytique  de  la  Chaleur,  Discours 
Preliminaire. 

613.  It  is  certainly  true  that  all  physical  phenomena  are 
subject  to  strictly  mathematical  conditions,  and  mathemati- 
cal processes  are  unassailable  in  themselves.     The  trouble  arises 
from  the  data  employed.    Most  phenomena  are  so  highly  com- 
plex that  one  can  never  be  quite  sure  that  he  is  dealing  with  all 
the  factors  until  the  experiment  proves  it.    So  that  experiment 
is  rather  the  criterion  of  mathematical  conclusions  and  must  lead 
the  way. — DOLBEAR,  A.  E. 

Matter,  Ether,  Motion  (Boston,  1894),  P-  89. 


90  MEMORABILIA   MATHEMATICA 

614.  Students  should  learn  to  study  at  an  early  stage  the 
great  works  of  the  great  masters  instead  of  making  their  minds 
sterile  through  the  everlasting  exercises  of  college,  which  are  of 
no  use  whatever,  except  to  produce  a  new  Arcadia  where  indo- 
lence is  veiled  under  the  form  of  useless  activity.  .  .  .  Hard 
study  on  the  great  models  has  ever  brought  out  the  strong; 
and  of  such  must  be  our  new  scientific  generation  if  it  is  to  be 
worthy  of  the  era  to  which  it  is  born  and  of  the  struggles  to 
which  it  is  destined. — BELTRAMI. 

Giornale  di  matematiche,    Vol.   11,   p.   153. 
[Young,  J.  W.] 

615.  The  history  of  mathematics  may  be  instructive  as  well  as 
agreeable ;  it  may  not  only  remind  us  of  what  we  have,  but  may 
also  teach  us  to  increase  our  store.  Says  De  Morgan,  "The  early 
history  of  the  mind  of  men  with  regards  to  mathematics  leads  us 
to  point  out  our  own  errors;  and  in  this  respect  it  is  well  to  pay 
attention  to  the  history  of  mathematics."    It  warns  us  against 
hasty  conclusions;  it  points  out  the  importance  of  a  good  nota- 
tion upon  the  progress  of  the  science;  it  discourages  excessive 
specialization  on  the  part  of  the  investigator,  by  showing  how 
apparently  distinct  branches  have  been  found  to  possess  unex- 
pected connecting  links;  it  saves  the  student  from  wasting  time 
and  energy  upon  problems  which  were,  perhaps,  solved  long 
since;  it  discourages  him  from  attacking  an  unsolved  problem 
by  the  same  method  which  has  led  other  mathematicians  to 
failure;  it  teaches  that  fortifications  can  be  taken  by  other  ways 
than  by  direct  attack,  that  when  repulsed  from  a  direct  assault 
it  is  well  to  reconnoitre  and  occupy  the  surrounding  ground  and 
to  discover  the  secret  paths  by  which  the  apparently  unconquer- 
able position  can  be  taken. — CAJORI,  F. 

History  of  Mathematics  (New  York,  1897), 
pp.  1-2. 

616.  The  history  of  mathematics  is  important  also  as  a  valu- 
able contribution  to  the  history  of  civilization.     Human  prog- 
ress is  closely  identified  with  scientific  thought.    Mathematical 
and  physical  researches  are  a  reliable  record  of  intellectual  pro- 
gress.— CAJORI,  F. 

History  of  Mathematics  (New  York,  1897), 
p.  4- 


STUDY   AND    RESEARCH   IN   MATHEMATICS  91 

617.  It  would  be  rash  to  say  that  nothing  remains  for  dis- 
covery or  improvement  even  in  elementary  mathematics,  but  it 
may  be  safely  asserted  that  the  ground  has  been  so  long  and  so 
thoroughly  explored  as  to  hold  out  little  hope  of  profitable  re- 
turn for  a  casual  adventurer. — TODHUNTER,  ISAAC. 

Private  Study  of  Mathematics;  Conflict  of 
Studies  and  other  Essays  (London,  1878], 
p.  73. 

618.  We  do  not  live  in  a  time  when  knowledge  can  be  ex- 
tended along  a  pathway  smooth  and  free  from  obstacles,  as  at 
the  time  of  the  discovery  of  the  infinitesimal  calculus,  and  in  a 
measure  also  when  in  the  development  of  protective  geometry 
obstacles  were  suddenly  removed  which,  having  hemmed  prog- 
ress for  a  long  time,  permitted  a  stream  of  investigators  to  pour 
in  upon  virgin  soil.    There  is  no  longer  any  browsing  along  the 
beaten  paths;  and  into  the  primeval  forest  only  those  may  ven- 
ture who  are  equipped  with  the  sharpest  tools. — BURKHARDT,  H. 

Mathematisches  und  wissenschaftliches  Den- 
ken;  Jahresbericht  der  Deutschen  Mathema- 
tiker  Vereinigung,  Bd.  11,  p.  55. 

619.  Though  we  must  not  without  further  consideration  con- 
demn a  body  of  reasoning  merely  because  it  is  easy,  neverthe- 
less we  must  not  allow  ourselves  to  be  lured  on  merely  by  easi- 
ness; and  we  should  take  care  that  every  problem  which  we 
choose  for  attack,  whether  it  be  easy  or  difficult,  shall  have  a 
useful  purpose,  that  it  shall  contribute  in  some  measure  to  the 
up-building  of  the  great  edifice. — SEGRE,  CORRADI. 

Some  Recent  Tendencies  in  Geometric  Investi- 
gation; Rivista  di  Matematica  (1891},  p.  68. 
Bulletin  American  Mathematical  Society, 
1904,  P-  465.  [Young,  J.  W.]. 

620.  No  mathematician  now-a-days  sets  any  store  on  the 
discovery  of  isolated  theorems,  except  as  affording  hints  of  an 
unsuspected  new  sphere  of  thought,  like  meteorites  detached 
from  some  undiscovered  planetary  orb  of  speculation. 

SYLVESTER,  J.  J. 

Notes  to  the  Exeter  Association  Address; 
Collected  Mathematical  Papers  (Cambridge, 
1908),  Vol.  2,  p.  715. 


92  MEMORABILIA    MATHEMATICA 

621.  Isolated,  so-called  "pretty  theorems"  have  even  less 
value  in  the  eyes  of  a  modern  mathematician  than  the  discovery 
of  a  new  "pretty  flower"  has  to  the  scientific  botanist,  though 
the  layman  finds  in  these  the  chief  charm  of  the  respective 
sciences. — HANKEL,  HERMANN. 

Die   Entwickelung   der   Mathematik   in   den 
letzten  Jahrhunderten  (Tubingen,  1884),  p.  15. 

622.  It  is,  so  to  speak,  a  scientific  tact,  which  must  guide 
mathematicians  in  their  investigations,  and  guard  them  from 
spending  their  forces  on  scientifically  worthless  problems  and 
abstruse  realms,  a  tact  which  is  closely  related  to  esthetic  tact  and 
which  is  the  only  thing  in  our  science  which  cannot  be  taught 
or  acquired,  and  is  yet  the  indispensable  endowment  of  every 
mathematician. — HANKEL,  HERMANN. 

Die  Entwickelung  der  Mathematik  in  den 
letzten  Jahrhunderten  (Tubingen,  1884),  P-  21. 

623.  The  mathematician  requires  tact  and  good  taste  at  every 
step  of  his  work,  and  he  has  to  learn  to  trust  to  his  own  instinct 
to  distinguish  between  what  is  really  worthy  of  his  efforts  and 
what  is  not;  he  must  take  care  not  to  be  the  slave  of  his  symbols, 
but  always  to  have  before  his  mind  the  realities  which  they 
merely  serve  to  express.    For  these  and  other  reasons  it  seems  to 
me  of  the  highest  importance  that  a  mathematician  should  be 
trained  in  no  narrow  school;  a  wide  course  of  reading  in  the  first 
few  years  of  his  mathematical  study  cannot  fail  to  influence  for 
good  the  character  of  the  whole  of  his  subsequent  work. 

GLAISHER,  J.  W.  L. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science,  Section  A,  (1890); 
Nature,  Vol.  42,  p.  467. 

624.  As  long  as  a  branch  of  science  offers  an  abundance  of 
problems,  so  long  it  is  alive;  a  lack  of  problems  foreshadows  ex- 
tinction or  the  cessation  of  independent  development. 

HILBERT,    D. 

Mathematical  Problems;  Bulletin  American 
Mathematical  Society,  Vol.  8,  p.  4®8. 


STUDY   AND    RESEARCH    IN   MATHEMATICS  93 

625.  In  mathematics  as  in  other  fields,  to  find  one  self  lost  in 
wonder  at  some  manifestation  is  frequently  the  half  of  a  new 
discovery. — DIRICHLET,  P.  G.  L. 

Werke,  Bd.  2  (Berlin,  1897),  p.  283. 

626.  The  student  of  mathematics  often  finds  it  hard  to  throw 
off  the  uncomfortable  feeling  that  his  science,  in  the  person  of 
his  pencil,  surpasses  him  in  intelligence, — an  impression  which 
the  great  Euler  confessed  he  often  could  not  get  rid  of.    This 
feeling  finds  a  sort  of  justification  when  we  reflect  that  the  ma- 
jority of  the  ideas  we  deal  with  were  conceived  by  others,  often 
centuries  ago.    In  a  great  measure  it  is  really  the  intelligence 
of  other  people  that  confronts  us  in  science. — MACH,  ERNST. 

Popular  Scientific  Lectures  (Chicago,  1910), 
p.  196. 

627.  It  is  probably  this  fact  [referring  to  the  circumstance 
that  the  problems  of  the  parallel  axiom,  the  squaring  of  the 
circle,  the  solution  of  the  equation  of  the  fifth  degree,  have 
finally  found  fully  satisfactory  and  rigorous  solutions]  along  with 
other  philosophical  reasons  that  gives  rise  to  the  conviction 
(which  every  mathematician  shares,  but  which  no  one  has  yet 
supported  by  proof)  that  every  definite  mathematical  problem 
must  necessarily  be  susceptible  of  an  exact  settlement,  either  in 
the  form  of  an  actual  answer  to  the  question  asked,  or  by  the 
proof  of  the  impossibility  of  its  solution  and  therewith  the  nec- 
essary failure  of  all  attempts.  .  .  .  This  conviction  of  the  solva- 
bility of  every  mathematical  problem  is  a  powerful  incentive  to 
the  worker.    We  hear  within  us  the  perpetual  call :  There  is  the 
problem.    Seek  its  solution.    You  can  find  it  by  pure  reason,  for 
in  mathematics  there  is  no  ignorabimus. — HILBERT,  D. 

Mathematical   Problems;   Bulletin   American 
Mathematical  Society,  Vol.  8,  pp.  444~44^- 

628.  He  who  seeks  for  methods  without  having  a  definite 
problem  in  mind  seeks  for  the  most  part  in  vain. — HILBERT,  D. 

Mathematical   Problems;   Bulletin   American 
Mathematical   Society,    Vol.   8,    p.   444- 

629.  A  mathematical  problem  should  be  difficult  in  order  to 
entice  us,  yet  not  completely  inaccessible,  lest  it  mock  at  our 


94  MEMORABILIA    MATHEMATICA 

efforts.  It  should  be  to  us  a  guide  post  on  the  mazy  paths  to 
hidden  truths,  and  ultimately  a  reminder  of  our  pleasure  in  the 
successful  solution. — HILBERT,  D. 

Mathematical  Problems;  Bulletin  American 
Mathematical  Society,  Vol.  8,  p.  438. 

630.  The  great  mathematicians  have  acted  on  the  principle 
"Divinez  avant  de  demontrer,"   and  it  is  certainly  true  that 
almost  all  important  discoveries  are  made  in  this  fashion. 

KASNER,  EDWARD. 

The  Present  Problems  in  Geometry;  Bulletin 
American  Mathematical  Society,  Vol.  11, 
p.  285. 

631.  "Divide  et  impera"  is  as  true  in  algebra  as  in  state- 
craft; but  no  less  true  and  even  more  fertile  is  the  maxim  "auge 
et  impera."    The  more  to  do  or  to  prove,  the  easier  the  doing  or 
the  proof. — SYLVESTER,  J.  J. 

Proof  of  the  Fundamental  Theorem  of  Inva- 
riants; Philosophic  Magazine  (1878),  p.  186; 
Collected  Mathematical  Papers,  Vol.  8, 
p.  126. 

632.  As  in  the  domains  of  practical  life  so  likewise  in  science 
there  has  come  about  a  division  of  labor.    The  individual  can 
no  longer  control  the  whole  field  of  mathematics:  it  is  only 
possible  for  him  to  master  separate  parts  of  it  in  such  a  manner 
as  to  enable  him  to  extend  the  boundaries  of  knowledge  by 
creative  research. — LAMPE,  E. 

Die  reine  Mathematik  in  den  Jahren  1884- 
1899,  p.  10. 

633.  With  the  extension  of  mathematical  knowledge  will  it 
not  finally  become  impossible  for  the  single  investigator  to 
embrace  all  departments  of  this  knowledge?    In  answer  let 
me  point  out  how  thoroughly  it  is  ingrained  in  mathematical 
science  that  every  real  advance  goes  hand  hi  hand  with  the 
invention  of  sharper  tools  and  simpler  methods  which  at  the 
same  time  assist  in  understanding  earlier  theories  and  to  cast 
aside  some  more  complicated  developments.     It  is  therefore 


STUDY   AND    RESEARCH    IN   MATHEMATICS  95 

possible  for  the  individual  investigator,  when  he  makes  these 
sharper  tools  and  simpler  methods  his  own,  to  find  his  way  more 
easily  in  the  various  branches  of  mathematics  than  is  possible 
in  any  other  science. — HILBERT,  D. 

Mathematical  Problems;  Bulletin  American 
Mathematical  Society,  Vol.  8,  p.  4?9. 

634.  It  would  seem  at  first  sight  as  if  the  rapid  expansion  of 
the  region  of  mathematics  must  be  a  source  of  danger  to  its 
future  progress.  Not  only  does  the  area  widen  but  the  subjects 
of  study  increase  rapidly  in  number,  and  the  work  of  the  mathe- 
matician tends  to  become  more  and  more  specialized.  It  is,  of 
course,  merely  a  brilliant  exaggeration  to  say  that  no  mathe- 
matician is  able  to  understand  the  work  of  any  other  mathe- 
matician, but  it  is  certainly  true  that  it  is  daily  becoming  more 
and  more  difficult  for  a  mathematician  to  keep  himself  ac- 
quainted, even  in  a  general  way,  with  the  progress  of  any  of  the 
branches  of  mathematics  except  those  which  form  the  field  of  his 
own  labours.  I  believe,  however,  that  the  increasing  extent  of 
the  territory  of  mathematics  will  always  be  counteracted  by 
increased  facilities  in  the  means  of  communication.  Additional 
knowledge  opens  to  us  new  principles  and  methods  which  may 
conduct  us  with  the  greatest  ease  to  results  which  previously 
were  most  difficult  of  access;  and  improvements  in  notation  may 
exercise  the  most  powerful  effects  both  in  the  simplification  and 
accessibility  of  a  subject.  It  rests  with  the  worker  in  mathe- 
matics not  only  to  explore  new  truths,  but  to  devise  the  lan- 
guage by  which  they  may  be  discovered  and  expressed;  and  the 
genius  of  a  great  mathematician  displays  itself  no  less  in  the 
notation  he  invents  for  deciphering  his  subject  than  in  the 
results  attained.  ...  I  have  great  faith  in  the  power  of  well- 
chosen  notation  to  simplify  complicated  theories  and  to  bring 
remote  ones  near  and  I  think  it  is  safe  to  predict  that  the  in- 
creased knowledge  of  principles  and  the  resulting  improvements 
in  the  symbolic  language  of  mathematics  will  always  enable  us 
to  grapple  satisfactorily  with  the  difficulties  arising  from  the 
mere  extent  of  the  subject. — GLAISHER,  J.  W.  L. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science,  Section  A.}  (1890), 
Nature,  Vol.  42,  p.  466. 


96  MEMORABILIA   MATHEMATICA 

635.  Quite  distinct  from  the  theoretical  question  of  the  man- 
ner in  which  mathematics  will  rescue  itself  from  the  perils  to 
which  it  is  exposed  by  its  own  prolific  nature  is  the  practical 
problem  of  finding  means  of  rendering  available  for  the  student 
the  results  which  have  been  already  accumulated,  and  making  it 
possible  for  the  learner  to  obtain  some  idea  of  the  present  state 
of  the  various  departments  of  mathematics.  .  .  .  The  great 
mass  of  mathematical  literature  will  be  always  contained  in 
Journals  and  Transactions,  but  there  is  no  reason  why  it  should 
not  be  rendered  far  more  useful  and  accessible  than  at  present 
by  means  of  treatises  or  higher  text-books.    The  whole  science 
suffers  from  want  of  avenues  of  approach,  and  many  beautiful 
branches  of  mathematics  are  regarded  as  difficult  and  technical 
merely  because  they  are  not  easily  accessible.  ...  I  feel  very 
strongly  that  any  introduction  to  a  new  subject  written  by  a 
competent  person  confers  a  real  benefit  on  the  whole  science. 
The  number  of  excellent  text-books  of  an  elementary  kind 
that  are  published  in  this  country  makes  it  all  the  more  to  be 
regretted  that  we  have  so  few  that  are  intended  for  the  ad- 
vanced student.    As  an  example  of  the  higher  kind  of  text-book, 
the  want  of  which  is  so  badly  felt  in  many  subjects,  I  may  men- 
tion the  second  part  of  Prof.  Chrystal's  "Algebra"  published 
last  year,  which  in  a  small  compass  gives  a  great  mass  of  valu- 
able and  fundamental  knowledge  that  has  hitherto  been  beyond 
the  reach  of  an  ordinary  student,  though  in  reality  lying  so  close 
at  hand.    I  may  add  that  hi  any  treatise  or  higher  text-book  it 
is  always  desirable  that  references  to  the  original  memoirs 
should  be  given,  and,  if  possible,  short  historic  notices  also. 
I  am  sure  that  no  subject  loses  more  than  mathematics  by  any 

attempt  to  dissociate  it  from  its  history. — GLAISHER,  J.  W.  L. 
Presidential  Address  British  Association  for 
the  Advancement  of  Science,  Section  A  (1890); 
Nature,  Vol.  42,  p.  466. 

636.  Tne  more  a  science  advances,  the  more  will  it  be  possible 
to  understand  immediately  results  which  formerly  could  be 
demonstrated  only  by  means  of  lengthy  intermediate  consid- 
erations: a  mathematical  subject  cannot  be  considered  as  finally 
completed  until  this  end  has  been  attained.  —  GORD  AN,  PAUL. 

Formensystem  bindrer  Formen  (Leipzig,  1875), 
p.  2. 


STUDY   AND    RESEARCH    IN   MATHEMATICS  97 

637.  An  old  French  geometer  used  to  say  that  a  mathematical 
theory  was  never  to  be  considered  complete  till  you  had  made  it 
so  clear  that  you  could  explain  it  to  the  first  man  you  met  in  the 
street. — SMITH,  H.  J.  S. 

Nature,  Vol.  8  (1873),  p.  452. 

638.  In  order  to  comprehend  and  fully  control  arithmetical 
concepts  and  methods  of  proof,  a  high  degree  of  abstraction  is 
necessary,  and  this  condition  has  at  times  been  charged  against 
arithmetic  as  a  fault.    I  am  of  the  opinion  that  all  other  fields  of 
knowledge  require  at  least  an  equally  high  degree  of  abstraction 
as  mathematics, — provided,  that  in  these  fields  the  foundations 
are  also  everywhere  examined  with  the  rigour  and  completeness 
which  is  actually  necessary. — HILBERT,  D. 

Die  Theorie  der  algebraischen  Zahlkorper, 
Vorwort;  Jahresbericht  der  Deutschen  Mathe- 
matiker  Vereinigung,  Bd.  4- 

639.  The  anxious  precision  of  modern  mathematics  is  neces- 
sary for  accuracy,  ...  it  is  necessary  for  research.    It  makes 
for  clearness  of  thought  and  for  fertility  in  trying  new  combina- 
tions of  ideas.    When  the  initial  statements  are  vague  and  slip- 
shod, at  every  subsequent  stage  of  thought,  common  sense  has 
to  step  in  to  limit  applications  and  to  explain  meanings.    Now  hi 
creative  thought  common  sense  is  a  bad  master.     Its  sole 
criterion  for  judgment  is  that  the  new  ideas  shall  look  like  the 
old  ones,  in  other  words  it  can  only  act  by  suppressing  original- 
ity.— WHITEHEAD,  A.  N. 

Introduction  to  Mathematics  (New  York, 
1911),  p.  157. 

640.  Mathematicians  attach  great  importance  to  the  elegance 
of  their  methods  and  their  results.    This  is  not  pure  dilettantism. 
What  is  it  indeed  that  gives  us  the  feeling  of  elegance  in  a  solu- 
tion, in  a  demonstration?   It  is  the  harmony  of  the  diverse  parts, 
their  symmetry,  their  happy  balance;  in  a  word  it  is  all  that 
introduces  order,  all  that  gives  unity,  that  permits  us  to  see 
clearly  and  to  comprehend  at  once  both  the  ensemble  and  the 
details.    But  this  is  exactly  what  yields  great  results,  in  fact  the 
more  we  see  this  aggregate  clearly  and  at  a  single  glance,  the 
better  we  perceive  its  analogies  with  other  neighboring  objects, 


98  MEMORABILIA   MATHEMATICA 

consequently  the  more  chances  we  have  of  divining  the  possible 
generalizations.  Elegance  may  produce  the  feeling  of  the  un- 
foreseen by  the  unexpected  meeting  of  objects  we  are  not  ac- 
customed to  bring  together;  there  again  it  is  fruitful,  since  it 
thus  unveils  for  us  kinships  before  unrecognized.  It  is  fruitful 
even  when  it  results  only  from  the  contrast  between  the  sim- 
plicity of  the  means  and  the  complexity  of  the  problem  set;  it 
makes  us  then  think  of  the  reason  for  this  contrast  and  very 
often  makes  us  see  that  chance  is  not  the  reason;  that  it  is  to  be 
found  in  some  unexpected  law.  In  a  word,  the  feeling  of  mathe- 
matical elegance  is  only  the  satisfaction  due  to  any  adaptation 
of  the  solution  to  the  needs  of  our  mind,  and  it  is  because  of  this 
very  adaptation  that  this  solution  can  be  for  us  an  instrument. 
Consequently  this  esthetic  satisfaction  is  bound  up  with  the 
economy  of  thought. — POINCARE,  H. 

The  Future  of  Mathematics;  Monist,  Vol.  20, 

p.  80.    [Halsted]. 

641.  The  importance  of  a  result  is  largely  relative,  is  judged 
differently  by  different  men,  and  changes  with  the  times  and 
circumstances.    It  has  often  happened  that  great  importance 
has  been  attached  to  a  problem  merely  on  account  of  the  diffi- 
culties which  it  presented;  and  indeed  if  for  its  solution  it  has 
been  necessary  to  invent  new  methods,  noteworthy  artifices, 
etc.,  the  science  has  gained  more  perhaps  through  these  than 
through  the  final  result.    In  general  we  may  call  important  all 
investigations  relating  to  things  which  in  themselves  are  im- 
portant; all  those  which  have  a  large  degree  of  generality,  or 
which  unite  under  a  single  point  of  view  subjects  apparently 
distinct,  simplifying  and  elucidating  them;  all  those  which  lead 
to  results  that  promise  to  be  the  source  of  numerous  conse- 
quences; etc. — SEGRE,  CORRADI. 

Some  Recent  Tendencies  in  Geometric  Investi- 
gations. Rivista  di  Matematica,  Vol.  l,p.  44- 
Bulletin  American  Mathematical  Society,  1904, 
p.  444-  [Young,  J.  W.]. 

642.  Geometric  writings  are  not  rare  in  which  one  would 
seek  in  vain  for  an  idea  at  all  novel,  for  a  result  which  sooner  or 
later  might  be  of  service,  for  anything  in  fact  which  might  be 


STUDY   AND    RESEARCH    IN   MATHEMATICS  99 

destined  to  survive  in  the  science;  and  one  finds  instead  treatises 
on  trivial  problems  or  investigations  on  special  forms  which 
have  absolutely  no  use,  no  importance,  which  have  their  origin 
not  in  the  science  itself  but  in  the  caprice  of  the  author;  or  one 
finds  applications  of  known  methods  which  have  already  been 
made  thousands  of  times;  or  generalizations  from  known  results 
which  are  so  easily  made  that  the  knowledge  of  the  latter  suffices 
to  give  at  once  the  former.  Now  such  work  is  not  merely  use- 
less; it  is  actually  harmful  because  it  produces  a  real  incumbrance 
in  the  science  and  an  embarrassment  for  the  more  serious  inves- 
tigators; and  because  often  it  crowds  out  certain  lines  of  thought 
which  might  well  have  deserved  to  be  studied. 

SEGRE,  CORRADI. 

On  some  Recent  Tendencies  in  Geometric  Inves- 
tigations; Rivista  di  Matematica,  1891,  p.  43. 
Bulletin  American  Mathematical  Society,  1904, 
p.  443  [Young,  J.  W.I 

643.  A  student  who  wishes  now-a-days  to  study  geometry  by 
dividing  it  sharply  from  analysis,  without  taking  account  of  the 
progress  which  the  latter  has  made  and  is  making,  that  student 
no  matter  how  great  his  genius,  will  never  be  a  whole  geometer. 
He  will  not  possess  those  powerful  instruments  of  research  which 
modern  analysis  puts  into  the  hands  of  modern  geometry. 
He  will  remain  ignorant  of  many  geometrical  results  which  are 
to  be  found,  perhaps  implicitly,  in  the  writings  of  the  analyst. 
And  not  only  will  he  be  unable  to  use  them  in  his  own  researches, 
but  he  will  probably  toil  to  discover  them  himself,  and,  as  hap- 
pens very  often,  he  will  publish  them  as  new,  when  really  he  has 
only  rediscovered  them. — SEGRE,  CORRADI. 

On  some  recent  Tendencies  in  Geometrical  In- 
vestigations; Rivista  di  Matematica,  1891,  p.  43. 
Bulletin  American  Mathematical  Society,  1904, 
p.  443  [Young,  J.  W.]. 

644.  Research  may  start  from  definite  problems  whose  im- 
portance it  recognizes  and  whose  solution  is  sought  more  or  less 
directly  by  all  forces.    But  equally  legitimate  is  the  other  method 
of  research  which  only  selects  the  field  of  its  activity  and,  con- 
trary to  the  first  method,  freely  reconnoitres  in  the  search  for 
problems  which  are  capable  of  solution.    Different  individuals 


100  MEMORABILIA   MATHEMATICA 

will  hold  different  views  as  to  the  relative  value  of  these  two 
methods.  If  the  first  method  leads  to  greater  penetration  it  is 
also  easily  exposed  to  the  danger  of  unproductivity.  To  the 
second  method  we  owe  the  acquisition  of  large  and  new  fields, 
in  which  the  details  of  many  things  remain  to  be  determined  and 
explored  by  the  first  method. — CLEBSCH,  A. 

Zum  Geddchtniss  an  Julius  Plucker;  Gdttinger 
Abhandlungen,  16,  1871,  Mathematische 
Classe,  p.  6. 

645.  During  a  conversation  with  the  writer  in  the  last  weeks  of 
his  life,  Sylvester  remarked  as  curious  that  notwithstanding 
he  had  always  considered  the  bent  of  his  mind  to  be  rather 
analytical  than  geometrical,  he  found  in  nearly  every  case  that 
the  solution  of  an  analytical  problem  turned  upon  some  quite 
simple  geometrical  notion,  and  that  he  was  never  satisfied  until 
he  could  present  the  argument  in  geometrical  language. 

MACMAHON,  P.  A. 

Proceedings  London  Royal  Society,  Vol. 
68,  p.  17. 

646.  The  origin  of  a  science  is  usually  to  be  sought  for  not  in 
any  systematic  treatise,  but  in  the  investigation  and  solution  of 
some  particular  problem.     This  is  especially  the  case  in  the 
ordinary  history  of  the  great  improvements  in  any  department 
of  mathematical  science.     Some    problem,   mathematical    or 
physical,  is  proposed,  which  is  found  to  be  insoluble  by  known 
methods.    This  condition  of  insolubility  may  arise  from  one  of 
two  causes:  Either  there  exists  no  machinery  powerful  enough 
to  effect  the  required  reduction,  or  the  workmen  are  not  suffi- 
ciently expert  to  employ  their  tools  in  the  performance  of  an 
entirely  new  piece  of  work.    The  problem  proposed  is,  however, 
finally  solved,  and  in  its  solution  some  new  principle,  or  new 
application  of  old  principles,  is  necessarily  introduced.     If  a 
principle  is  brought  to  light  it  is  soon  found  that  in  its  applica- 
tion it  is  not  necessarily  limited  to  the  particular  question  which 
occasioned  its  discovery,  and  it  is  then  stated  in  an  abstract  form 
and  applied  to  problems  of  gradually  increasing  generality. 

Other  principles,  similar  in  their  nature,  are  added,  and  the 
original  principle  itself  receives  such  modifications  and  exten- 


STUDY   AND    RESEARCH    IN   MATHEMATICS  101 

sions  as  are  from  time  to  time  deemed  necessary.  The  same  is 
true  of  new  applications  of  old  principles;  the  application  is 
first  thought  to  be  merely  confined  to  a  particular  problem,  but 
it  is  soon  recognized  that  this  problem  is  but  one,  and  gener- 
ally a  very  simple  one,  out  of  a  large  class,  to  which  the  same 
process  of  investigation  and  solution  are  applicable.  The  result 
in  both  of  these  cases  is  the  same.  A  time  comes  when  these 
several  problems,  solutions,  and  principles  are  grouped  together 
and  found  to  produce  an  entirely  new  and  consistent  method; 
a  nomenclature  and  uniform  system  of  notation  is  adopted,  and 
the  principles  of  the  new  method  become  entitled  to  rank  as  a 
distinct  science. — CRAIG,  THOMAS. 

A  Treatise  on  Projection,  Preface,  U.  S.  Coast 
and  Geodetic  Survey,  Treasury  Department 
Document,  No.  61. 

647.  The  aim  of  research  is  the  discovery  of  the  equations 
which  subsist  between  the  elements  of  phenomena. 

MACH,  ERNST. 

Popular  Scientific  Lectures  (Chicago,  1910), 
p.  205. 

648.  Let  him  [the  author]  be  permitted  also  in  all  humility  to 
add  .  .  .  that  in  consequence  of  the  large  arrears  of  algebraical 
and  arithmetical  speculations  waiting  in  his  mind  their  turn  to 
be  called  into  outward  existence,  he  is  driven  to  the  alternative 
of  leaving  the  fruits  of  his  meditations  to  perish  (as  has  been  the 
fate  of  too  many  foregone  theories,  the  still-born  progeny  of 
his  brain,  now  forever  resolved  back  again  into  the  primordial 
matter  of  thought),  or  venturing  to  produce  from  time  to  time 
such  imperfect  sketches  as  the  present,  calculated  to  evoke  the 
mental  co-operation  of  his  readers,  in  whom  the  algebraical 
instinct  has  been  to  some  extent  developed,  rather  than  to 
satisfy  the  strict  demands  of  rigorously  systematic  exposition. 

SYLVESTER,  J.  J. 
Philosophic  Magazine  (1863),  p.  460. 

649.  In  other  branches  of  science,  where  quick  publication 
seems  to  be  so  much  desired,  there  may  possibly  be  some  excuse 
for  giving  to  the  world  slovenly  or  ill-digested  work,  but  there 
is  no  such  excuse  in  mathematics.    The  form  ought  to  be  as 


LIBRARY 

UNIVERSITY  OF  CALIFORNIA 
SANTA  BARBARA 


102  MEMORABILIA   MATHEMATICA 

perfect  as  the  substance,  and  the  demonstrations  as  rigorous 
as  those  of  Euclid.  The  mathematician  has  to  deal  with  the 
most  exact  facts  of  Nature,  and  he  should  spare  no  effort  to 
render  his  interpretation  worthy  of  his  subject,  and  to  give  to 
his  work  its  highest  degree  of  perfection.  "Pauca  sed  matura" 
was  Gauss's  motto. — GLAISHER,  J.  W.  L. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science,  Section  A,  (1890); 
Nature,  Vol.  42,  p.  467. 

650.  It  is  the  man  not  the  method  that  solves  the  problem. 

MASCHKE,  H. 

Present  Problems  of  Algebra  and  Analysis; 
Congress  of  Arts  and  Sciences  (New  York  and 
Boston,  1905},  Vol.  1,  p.  530. 

651.  Today  it  is  no  longer  questioned  that  the  principles  of 
the  analysts  are  the  more  far-reaching.    Indeed,  the  synthesists 
lack  two  things  in  order  to  engage  in  a  general  theory  of  alge- 
braic configurations:  these  are  on  the  one  hand  a  definition  of 
imaginary  elements,  on  the  other  an  interpretation  of  general 
algebraic  concepts.     Both  of  these  have  subsequently  been 
developed  in  synthetic  form,  but  to  do  this  the  essential  principle 
of  synthetic  geometry  had  to  be  set  aside.    This  principle  which 
manifests  itself  so  brilliantly  in  the  theory  of  linear  forms  and  the 
forms  of  the  second  degree,  is  the  possibility  of  immediate 
proof  by  means  of  visualized  constructions. — KLEIN,  FELIX. 

Riemannsche  Flachen  (Leipzig,  1906),  Bd.  1, 
p.  234. 

652.  Abstruse  mathematical  researches  .  .  .  are  .  .  .  often 
abused  for  having  no  obvious  physical  application.    The  fact 
is  that  the  most  useful  parts  of  science  have  been  investigated 
for  the  sake  of  truth,  and  not  for  their  usefulness.    A  new  branch 
of  mathematics,  which  has  sprung  up  in  the  last  twenty  years, 
was  denounced  by  the  Astronomer  Royal  before  the  University 
of  Cambridge  as  doomed  to  be  forgotten,  on  account  of  its 
uselessness.    Now  it  turns  out  that  the  reason  why  we  cannot  go 
further  in  our  investigations  of  molecular  action  is  that  we  do  not 
know  enough  of  this  branch  of  mathematics. — CLIFFORD,  W.  K. 

Conditions  of  Mental  Development;  Lectures 
and  Essays  (London,  1901),  Vol.  1,  p.  115. 


STUDY   AND    RESEARCH    IN    MATHEMATICS  103 

653.  In  geometry,  as  in  most  sciences,  it  is  very  rare  that  an 
isolated  proposition  is  of  immediate  utility.    But  the  theories 
most  powerful  in  practice  are  formed  of  propositions  which 
curiosity  alone  brought  to  light,  and  which  long  remained  use- 
less without  its  being  able  to  divine  in  what  way  they  should 
one  day  cease  to  be  so.    In  this  sense  it  may  be  said,  that  in  real 
science,  no  theory,  no  research,  is  in  effect  useless. — VOLTAIRE. 

A  Philosophical  Dictionary,  Article  "Geome- 
try"; (Boston,  1881),  Vol.  1,  p.  374. 

654.  Scientific  subjects  do  not  progress  necessarily  on  the 
lines  of  direct  usefulness.     Very  many  applications  of  the 
theories  of  pure  mathematics  have  come  many  years,  sometimes 
centuries,  after  the  actual  discoveries  themselves.    The  weapons 
were  at  hand,  but  the  men  were  not  able  to  use  them. 

FORSYTH,  A.  R. 

Perry's   Teaching   of  Mathematics   (London, 
1902),  p.  35. 

655.  It  is  no  paradox  to  say  that  in  our  most  theoretical  moods 
we  may  be  nearest  to  our  most  practical  applications. 

WHITEHEAD,  A.  N. 

Introduction    to    Mathematics    (New    York), 
p.  100. 

656.  Although  with  the  majority  of  those  who  study  and 
practice   in   these   capacities   [engineers,   builders,   surveyors, 
geographers,  navigators,  hydrographers,  astronomers],  second- 
hand acquirements,  trite  formulas,  and  appropriate  tables  are 
sufficient  for  ordinary  purposes,  yet  these  trite  formulas  and 
familiar  rules  were  originally  or  gradually  deduced  from  the 
profound  investigations  of  the  most  gifted  minds,  from  the 
dawn  of  science  to  the  present  day.  .  .  .    The  further  develop- 
ments of  the  science,  with  its  possible  applications  to  larger 
purposes  of  human  utility  and  grander  theoretical  generaliza- 
tions, is  an  achievement  reserved  for  a  few  of  the  choicest  spirits, 
touched  from  time  to  time  by  Heaven  to  these  highest  issues. 
The  intellectual  world  is  filled  with  latent  and  undiscovered 
truth  as  the  material  world  is  filled  with  latent  electricity. 

EVERETT,  EDWARD. 

Orations  and  Speeches,  Vol.  8  (Boston,  1870), 
p.  513. 


104  MEMORABILIA    MATHEMATICA 

667.  If  we  view  mathematical  speculations  with  reference  to 
their  use,  it  appears  that  they  should  be  divided  into  two 
classes.  To  the  first  belong  those  which  furnish  some  marked 
advantage  either  to  common  life  or  to  some  art,  and  the  value 
of  such  is  usually  determined  by  the  magnitude  of  this  advan- 
tage. The  other  class  embraces  those  speculations  which, 
though  offering  no  direct  advantage,  are  nevertheless  valuable 
in  that  they  extend  the  boundaries  of  analysis  and  increase  our 
resources  and  skill.  Now  since  many  investigations,  from  which 
great  advantage  may  be  expected,  must  be  abandoned  solely 
because  of  the  imperfection  of  analysis,  no  small  value  should 
be  assigned  to  those  speculations  which  promise  to  enlarge  the 
field  of  anaylsis. — EULER. 

Novi  Comm.  Petr.,  Vol.  4,  Preface. 

658.  The  discovery  of  the  conic  sections,  attributed  to  Plato, 
first  threw  open  the  higher  species  of  form  to  the  contempla- 
tion of  geometers.    But  for  this  discovery,  which  was  probably 
regarded  in  Plato's  time  and  long  after  him,  as  the  unprofitable 
amusement  of  a  speculative  brain,  the  whole  course  of  practical 
philosophy  of  the  present  day,  of  the  science  of  astronomy,  of 
the  theory  of  projectiles,  of  the  art  of  navigation,  might  have 
run  in  a  different  channel;  and  the  greatest  discovery  that  has 
ever  been  made  in  the  history  of  the  world,  the  law  of  universal 
gravitation,  with  its  innumerable  direct  and  indirect  conse- 
quences and  applications  to  every  department  of  human  re- 
search and  industry,  might  never  to  this  hour  have  been  elicited. 

SYLVESTER,  J.  J. 

A  Probationary  Lecture  on  Geometry;  Col- 
lected Mathematical  Papers,  Vol.  2  (Cam- 
bridge, 1908},  p.  7. 

659.  No  more  impressive  warning  can  be  given  to  those  who 
would  confine  knowledge  and  research  to  what  is  apparently 
useful,  than  the  reflection  that  conic  sections  were  studied  for 
eighteen  hundred  years  merely  as  an  abstract  science,  without 
regard  to  any  utility  other  than  to  satisfy  the  craving  for  knowl- 
edge on  the  part  of  mathematicians,  and  that  then  at  the  end  of 
this  long  period  of  abstract  study,  they  were  found  to  be  the 


STUDY   AND    RESEARCH    IN    MATHEMATICS  105 

necessary  key  with  which  to  attain  the  knowledge  of  the  most 
important  laws  of  nature. — WHITEHEAD,  A.  N. 

Introduction    to    Mathematics    (New    York, 

York,  1911),  pp.  136-137. 

660.  The  Greeks  in  the  first  vigour  of  their  pursuit  of  mathe- 
matical truth,  at  the  time  of  Plato  and  soon  after,  had  by  no 
means  confined  themselves  to  those  propositions  which  had  a 
visible  bearing  on  the  phenomena  of  nature;  but  had  followed 
out  many  beautiful  trains  of  research  concerning  various  kinds 
of  figures,  for  the  sake  of  their  beauty  alone;  as  for  instance  in 
their  doctrine  of  Conic  Sections,  of  which  curves  they  had  dis- 
covered all  the  principal  properties.    But  it  is  curious  to  remark, 
that  these  investigations,  thus  pursued  at  first  as  mere  matters 
of  curiosity  and  intellectual  gratification,  were  destined,  two 
thousand  years  later,  to  play  a  very  important  part  in  estab- 
lishing that  system  of  celestial  motions  which  succeeded  the 
Platonic  scheme  of  cycles  and  epicycles.    If  the  properties  of 
conic  sections  had  not  been  demonstrated  by  the  Greeks  and 
thus  rendered  familiar  to  the  mathematicians  of  succeeding 
ages,  Kepler  would  probably  not  have  been  able  to  discover 
those  laws  respecting  the  orbits  and  motions  of  planets  which 
were  the  occasion  of  the  greatest  revolution  that  ever  happened 
in  the  history  of  science. — WHEWELL,  W. 

History  of  Scientific  Ideas,  Bk.  2,  chap.  14, 
sect.  3. 

661.  The  greatest  mathematicians,  as  Archimedes,  Newton, 
and  Gauss,  always  united  theory  and  applications  in  equal 
measure. — KLEIN,  FELIX. 

Elementarmathematik    vom    hdheren    Stand- 
punkte  aus  (Leipzig,  1909),  Bd.  2,  p.  892. 

662.  We  may  see  how  unexpectedly  recondite  parts  of  pure 
mathematics  may  bear  upon  physical  science,  by  calling  to 
mind  the  circumstance  that  Fresnel  obtained  one  of  the  most 
curious  confirmations  of  the  theory  (the  laws  of  Circular  Polari- 
zation by  reflection)  through  an  interpretation  of  an  algebraical 
expression,  which,  according  to  the  original  conventional  mean- 
ing of  the  symbols,  involved  an  impossible  quantity. 

WHEWELL,  W. 
History  of  Scientific  Ideas,  Bk.  2,  chap.  14,  sect.  8. 


106  MEMORABILIA    MATHEMATICA 

663.  A  great  department  of  thought  must  have  its  own  inner 
life,  however  transcendent  may  be  the  importance  of  its  rela- 
tions to  the  outside.    No  department  of  science,  least  of  all  one 
requiring  so  high  a  degree  of  mental  concentration  as  Mathe- 
matics, can  be  developed  entirely,  or  even  mainly,  with  a  view  to 
applications  outside  its  own  range.    The  increased  complexity 
and  specialisation  of  all  branches  of  knowledge  makes  it  true 
in  the  present,  however  it  may  have  been  in  former  times,  that 
important  advances  in  such  a  department  as  Mathematics  can 
be  expected  only  from  men  who  are  interested  in  the  subject  for 
its  own  sake,  and  who,  whilst  keeping  an  open  mind  for  sugges- 
tions from  outside,  allow  their  thought  to  range  freely  in  those 
lines  of  advance  which  are  indicated  by  the  present  state  of 
their  subject,  untrammelled  by  any  preoccupation  as  to  applica- 
tions to  other  departments  of  science.     Even  with  a  view  to 
applications,  if  Mathematics  is  to  be  adequately  equipped  for 
the  purpose  of  coping  with  the  intricate  problems  which  will  be 
presented  to  it  in  the  future  by  Physics,  Chemistry  and  other 
branches  of  physical  science,  many  of  these  problems  probably 
of  a  character  which  we  cannot  at  present  forecast,  it  is  essential 
that  Mathematics  should  be  allowed  to  develop  freely  on  its 

own  lines. — HOBSON,  E.  W. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science,  Section  A,  (1910); 
Nature,  Vol.  84,  p.  286. 

664.  To  emphasize  this  opinion  that  mathematicians  would 
be  unwise  to  accept  practical  issues  as  the  sole  guide  or  the 
chief  guide  in  the  current  of  their  investigations,  ...  let  me 
take  one  more  instance,  by  choosing  a  subject  in  which  the 
purely  mathematical  interest  is  deemed  supreme,  the  theory  of 
functions  of  a  complex  variable.    That  at  least  is  a  theory  in 
pure  mathematics,  initiated  in  that  region,  and  developed  in 
that  region;  it  is  built  up  in  scores  of  papers,  and  its  plan  cer- 
tainly has  not  been,  and  is  not  now,  dominated  or  guided  by 
considerations  of  applicability  to  natural  phenomena.    Yet  what 
has  turned  out  to  be  its  relation  to  practical  issues?   The  investi- 
gations of  Lagrange  and  others  upon  the  construction  of  maps 
appear  as  a  portion  of  the  general  property  of  conformal  repre- 
sentation; which  is  merely  the  general  geometrical  method  of 


STUDY   AND    RESEARCH    IN   MATHEMATICS  107 

regarding  functional  relations  in  that  theory.  Again,  the 
interesting  and  important  investigations  upon  discontinuous 
two-dimensional  fluid  motion  in  hydrodynamics,  made  in  the 
last  twenty  years,  can  all  be,  and  now  are  all,  I  believe,  deduced 
from  similar  considerations  by  interpreting  functional  relations 
between  complex  variables.  In  the  dynamics  of  a  rotating  heavy 
body,  the  only  substantial  extension  of  our  knowledge  since  the 
time  of  Lagrange  has  accrued  from  associating  the  general 
properties  of  functions  with  the  discussion  of  the  equations  of 
motion.  Further,  under  the  title  of  conjugate  functions,  the 
theory  has  been  applied  to  various  questions  in  electrostatics, 
particularly  in  connection  with  condensors  and  electrometers. 
And,  lastly,  in  the  domain  of  physical  astronomy,  some  of  the 
most  conspicuous  advances  made  in  the  last  few  years  have 
been  achieved  by  introducing  into  the  discussion  the  ideas,  the 
principles,  the  methods,  and  the  results  of  the  theory  of  func- 
tions. .  .  .  the  refined  and  extremely  difficult  work  of  Poincare 
and  others  in  physical  astronomy  has  been  possible  only  by  the 
use  of  the  most  elaborate  developments  of  some  purely  mathe- 
matical subjects,  developments  which  were  made  without  a 
thought  of  such  applications. — FORSYTH,  A.  R. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science,  Section  A,  (1897); 
Nature,  Vol.  56,  p.  377. 


CHAPTER  VII 

MODERN   MATHEMATICS 

701.  Surely  this  is  the  golden  age  of  mathematics. 

PIERPONT,  JAMES. 

History  of  Mathematics  in  the  Nineteenth  Cen- 
tury; Congress  of  Arts  and  Sciences  (Boston 
and  New  York,  1905),  Vol.  1,  p.  493. 

702.  The  golden  age  of  mathematics — that  was  not  the  age  of 
Euclid,  it  is  ours.    Ours  is  the  age  when  no  less  than  six  inter- 
national congresses  have  been  held  in  the  course  of  nine  years. 
It  is  in  our  day  that  more  than  a  dozen  mathematical  societies 
contain  a  growing   membership  of  more  than   two  thousand 
men  representing  the  centers  of  scientific  light  throughout  the 
great  culture  nations  of  the  world.    It  is  in  our  time  that  over 
five  hundred  scientific  journals  are  each  devoted  in  part,  while 
more  than  two  score  others  are  devoted  exclusively,  to  the 
publication  of  mathematics.    It  is  in  our  time  that  the  Jahrbuch 
iiber  die  Fortschritte  der  Mathematik,  though  admitting  only 
condensed  abstracts  with  titles,  and  not  reporting  on  all  the 
journals,  has,  nevertheless,  grown  to  nearly  forty  huge  volumes 
in  as  many  years.    It  is  in  our  time  that  as  many  as  two  thou- 
sand books  and  memoirs  drop  from  the  mathematical  press  of 
the  world  in  a  single  year,  the  estimated  number  mounting  up  to 
fifty  thousand  in  the  last  generation.    Finally,  to  adduce  yet 
another  evidence  of  a  similar  kind,  it  requires  not  less  than 
seven  ponderous  tomes  of  the  forthcoming  Encyclopaedic  der 
Mathematischen  Wissenschaften  to  contain,  not  expositions,  not 
demonstrations,  but  merely  compact  reports  and  bibliographic 
notices  sketching  developments  that  have  taken  place  since  the 
beginning  of  the  nineteenth  century. — KEYSER,  C.  J. 

Lectures    on    Science,    Philosophy    and    Art 
(New  York,  1908),  p.  8. 

703.  I  have  said  that  mathematics  is  the  oldest  of  the  sciences; 
a  glance  at  its  more  recent  history  will  show  that  it  has  the 

108 


MODERN   MATHEMATICS  109 

energy  of  perpetual  youth.  The  output  of  contributions  to  the 
advance  of  the  science  during  the  last  century  and  more  has 
been  so  enormous  that  it  is  difficult  to  say  whether  pride  in  the 
greatness  of  achievement  in  this  subject,  or  despair  at  his 
inability  to  cope  with  the  multiplicity  of  its  detailed  develop- 
ments, should  be  the  dominant  feeling  of  the  mathematician. 
Few  people  outside  of  the  small  circle  of  mathematical  special- 
ists have  any  idea  of  the  vast  growth  of  mathematical  literature. 
The  Royal  Society  Catalogue  contains  a  list  of  nearly  thirty- 
nine  thousand  papers  on  subjects  of  Pure  Mathematics  alone, 
which  have  appeared  in  seven  hundred  serials  during  the  nine- 
teenth century.  This  represents  only  a  portion  of  the  total 
output,  the  very  large  number  of  treatises,  dissertations,  and 
monographs  published  during  the  century  being  omitted. 

HOBSON,  E.  W. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science,  Section  A,  (1910}; 
Nature,  Vol.  84,  p.  283. 

704.  Mathematics  is  one  of  the  oldest  of  the  sciences;  it  is 
also  one  of  the  most  active,  for  its  strength  is  the  vigour  of 
perpetual  youth. — FORSYTH,  A.  R. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science,  Section  A,  (1897); 
Nature,  Vol.  56,  p.  878. 

705.  The  nineteenth  century  which  prides  itself  upon  the 
invention  of  steam  and  evolution,  might  have  derived  a  more 
legitimate  title  to  fame  from  the  discovery  of  pure  mathe- 
matics.— RUSSELL,  BERTRAND. 

International  Monthly,  Vol.  4  (1901),  p.  83. 

706.  One  of  the  chiefest  triumphs  of  modern  mathematics 
consists  in  having  discovered  what  mathematics  really  is. 

RUSSELL,  BERTRAND. 
International  Monthly,  Vol.  4  (1901),  p.  84- 

707.  Modern  mathematics,  that  most  astounding  of  intel- 
lectual creations,  has  projected  the  mind's  eye  through  infinite 
time  and  the  mind's  hand  into  boundless  space. — BUTLER,  N.  M. 

The  Meaning  of  Education  and  other  Essays 
and   Addresses    (New    York,    1905),   p.   44- 


110  MEMORABILIA  MATHEMATICA 

708.  The  extraordinary  development  of  mathematics  in  the 
last  century  is  quite  unparalleled  in  the  long  history  of  this  most 
ancient  of  sciences.    Not  only  have  those  branches  of  mathe- 
matics which  were  taken  over  from  the  eighteenth  century 
steadily  grown,  but  entirely  new  ones  have  sprung  up  in  almost 
bewildering  profusion,  and  many  of  them  have  promptly  as- 
sumed proportions  of  vast  extent. — PIEBPONT,  J. 

The  History  of  Mathematics  in  the  Nine- 
teenth Century;  Congress  of  Arts  and  Sciences 
(Boston  and  New  York,  1905),  Vol.  1,  p.  474. 

709.  The  Modern  Theory  of  Functions — that  stateliest  of 
all  the  pure  creations  of  the  human  intellect. — KEYSER,  C.  J. 

Lectures  on  Science,  Philosophy  and  Art  (New 
York,  1908),  p.  16. 

710.  If  a  mathematician  of  the  past,  an  Archimedes  or  even 
a  Descartes,  could  view  the  field  of  geometry  in  its  present 
condition,  the  first  feature  to  impress  him  would  be  its  lack  of 
concreteness.     There  are  whole  classes  of  geometric  theories 
which  proceed  not  only  without  models  and  diagrams,  but 
without  the  slightest  (apparent)  use  of  spatial  intuition.    In  the 
main  this  is  due,  to  the  power  of  the  analytic  instruments  of 
investigations  as  compared  with  the  purely  geometric. 

KASNER,  EDWARD. 

The  Present  Problems  in  Geometry;  Bulletin 
American  Mathematical  Society,  1905,  p.  285. 

711.  In  Euclid  each  proposition  stands  by  itself ;  its  connection 
with  others  is  never  indicated;  the  leading  ideas  contained  in 
its  proof  are  not  stated;  general  principles  do  not  exist.     In 
modern  methods,  on  the  other  hand,  the  greatest  importance 
is  attached  to  the  leading  thoughts  which  pervade  the  whole; 
and  general  principles,  which  bring  whole  groups  of  theorems 
under  one  aspect,  are  given  rather  than  separate  propositions. 
The  whole  tendency  is  toward  generalization.    A  straight  line  is 
considered  as  given  in  its  entirety,  extending  both  ways  to 
infinity,  while  Euclid  is  very  careful  never  to  admit  anything  but 
finite  quantities.    The  treatment  of  the  infinite  is  in  fact  another 


MODERN   MATHEMATICS  111 

fundamental  difference  between  the  two  methods.  Euclid 
avoids  it,  in  modern  mathematics  it  is  systematically  introduced, 
for  only  thus  is  generality  obtained. — CAYLEY,  ARTHUR. 

Encyclopedia  Britannica  (9th  edition),  Article 
"Geometry." 

712.  This  is  one  of  the  greatest  advantages  of  modern  geome- 
try over  the  ancient,  to  be  able,  through  the  consideration  of 
positive  and  negative  quantities,  to  include  in  a  single  enuncia- 
tion the  several  cases  which  the  same  theorem  may  present  by  a 
change  in  the  relative  position  of  the  different  parts  of  a  figure. 
Thus  in  our  day  the  nine  principal  problems  and  the  numerous 
particular  cases,  which  form  the  object  of  eighty-three  theorems 
in  the  two  books  De  sectione  determinate,  of  Appolonius  con- 
stitute only  one  problem  which  is  resolved  by  a  single  equation. 

'    CHASLES,  M. 
Histoire  de  la  Geometric,  chap.  1,  sect.  35. 

713.  Euclid  always  contemplates  a  straight  line  as  drawn 
between  two  definite  points,  and  is  very  careful  to  mention  when 
it  is  to  be  produced  beyond  this  segment.    He  never  thinks  of 
the  line  as  an  entity  given  once  for  all  as  a  whole.    This  careful 
definition  and  limitation,  so  as  to  exclude  an  infinity  not  im- 
mediately apparent  to  the  senses,  was  very  characteristic  of  the 
Greeks  in  all  their  many  activities.    It  is  enshrined  in  the  differ- 
ence between  Greek  architecture  and  Gothic  architecture,  and 
between  Greek  religion  and  modern  religion.    The  spire  of  a 
Gothic  cathedral  and  the  importance  of  the  unbounded  straight 
line  in  modern  Geometry  are  both  emblematic  of  the  transforma- 
tion of  the  modern  world. — WHITEHEAD,  A.  N. 

Introduction  to  Mathematics  (New  York,  1911), 
p.  119. 

714.  The  geometrical  problems  and  theorems  of  the  Greeks 
always  refer  to  definite,  oftentimes  to  rather  complicated  fig- 
ures.   Now  frequently  the  points  and  lines  of  such  a  figure  may 
assume  very  many  different  relative  positions;  each  of  these 
possible  cases  is  then  considered  separately.    On  the  contrary, 
present  day  mathematicians  generate  their  figures  one  from 
another,  and  are  accustomed  to  consider  them  subject  to  vari- 


112  MEMORABILIA   MATHEMATICA 

ation;  in  this  manner  they  unite  the  various  cases  and  com- 
bine them  as  much  as  possible  by  employing  negative  and 
imaginary  magnitudes.  For  example,  the  problems  which 
Appolonius  treats  in  his  two  books  De  sectione  rationis,  are 
solved  today  by  means  of  a  single,  universally  applicable  con- 
struction; Apollonius,  on  the  contrary,  separates  it  into  more 
than  eighty  different  cases  varying  only  in  position.  Thus,  as 
Hermann  Hankel  has  fittingly  remarked,  the  ancient  geometry 
sacrifices  to  a  seeming  simplicity  the  true  simplicity  which 
consists  in  the  unity  of  principles;  it  attained  a  trivial  sensual 
presentability  at  the  cost  of  the  recognition  of  the  relations  of 
geometric  forms  in  all  their  changes  and  in  all  the  variations  of 
their  sensually  presentable  positions. — REYE,  THEODORE. 

Die  synthetische  Geometric  im  Altertum  und  in 
der  Neuzeit;  Jahresbericht  derDeutschen  Mathe- 
matiker  Vereinigung,  Bd.  2,  pp.  846-347. 

715.  It  is  known  that  the  mathematics  prescribed  for  the 
high  school  [Gymnasien]  is  essentially  Euclidean,  while  it  is 
modern  mathematics,  the  theory  of  functions  and  the  infinitesi- 
mal calculus,  which  has  secured  for  us  an  insight  into  the  mech- 
anism and  laws  of  nature.  Euclidean  mathematics  is  indeed, 
a  prerequisite  for  the  theory  of  functions,  but  just  as  one, 
though  he  has  learned  the  inflections  of  Latin  nouns  and  verbs, 
will  not  thereby  be  enabled  to  read  a  Latin  author  much  less  to 
appreciate  the  beauties  of  a  Horace,  so  Euclidean  mathematics, 
that  is  the  mathematics  of  the  high  school,  is  unable  to  unlock 
nature  and  her  laws.  Euclidean  mathematics  assumes  the 
completeness  and  invariability  of  mathematical  forms;  these 
forms  it  describes  with  appropriate  accuracy  and  enumerates 
their  inherent  and  related  properties  with  perfect  clearness, 
order,  and  completeness,  that  is,  Euclidean  mathematics 
operates  on  forms  after  the  manner  that  anatomy  operates  on 
the  dead  body  and  its  members. 

On  the  other  hand,  the  mathematics  of  variable  magnitudes — 
function  theory  or  analysis — considers  mathematical  forms  in 
their  genesis.  By  writing  the  equation  of  the  parabola,  we 
express  its  law  of  generation,  the  law  according  to  which  the 
variable  point  moves.  The  path,  produced  before  the  eyes  of  the 


MODERN   MATHEMATICS  113 

student  by  a  point  moving  in  accordance  to  this  law,  is  the 
parabola. 

If,  then,  Euclidean  mathematics  treats  space  and  number 
forms  after  the  manner  in  which  anatomy  treats  the  dead  body, 
modern  mathematics  deals,  as  it  were,  with  the  living  body,  with 
growing  and  changing  forms,  and  thus  furnishes  an  insight,  not 
only  into  nature  as  she  is  and  appears,  but  also  into  nature  as 
she  generates  and  creates, — reveals  her  transition  steps  and  in 
so  doing  creates  a  mind  for  and  understanding  of  the  laws  of 
becoming.  Thus  modern  mathematics  bears  the  same  relation 
to  Euclidean  mathematics  that  physiology  or  biology  .  .  .  bears 
to  anatomy.  But  it  is  exactly  in  this  respect  that  our  view  of 
nature  is  so  far  above  that  of  the  ancients;  that  we  no  longer 
look  on  nature  as  a  quiescent  complete  whole,  which  compels 
admiration  by  its  sublimity  and  wealth  of  forms,  but  that  we 
conceive  of  her  as  a  vigorous  growing  organism,  unfolding  ac- 
cording to  definite,  as  delicate  as  far-reaching,  laws;  that  we  are 
able  to  lay  hold  of  the  permanent  amidst  the  transitory,  of  law 
amidst  fleeting  phenomena,  and  to  be  able  to  give  these  their 
simplest  and  truest  expression  through  the  mathematical 
formulas. — DILLMANN,  E. 

Die  Mathematik  die  Fackeltragerin  einer 
neuen  Zeit  (Stuttgart,  1889),  p.  87. 

716.  The  Excellence  of  Modern  Geometry  is  in  nothing  more 
evident,  than  in  those  full  and  adequate  Solutions  it  gives  to 
Problems;  representing  all  possible  Cases  in  one  view,  and  in  one 
general  Theorem  many  times  comprehending  whole  Sciences; 
which  deduced  at  length  into  Propositions,  and  demonstrated 
after  the  manner  of  the  Ancients,  might  well  become  the  sub- 
jects of  large  Treatises:  For  whatsoever  Theorem  solves  the 
most  complicated  Problem  of  the  kind,  does  with  a  due  Reduc- 
tion reach  all  the  subordinate  Cases. — HALLEY,  E. 

An  Instance  of  the  Excellence  of  Modern  Al- 
gebra, etc.;  Philosophical  Transactions,  1694, 
p.  960. 

717.  One  of  the  most  conspicuous  and  distinctive  features  of 
mathematical  thought  in  the  nineteenth  century  is  its  critical 


114  MEMORABILIA   MATHEMATICA 

spirit.  Beginning  with  the  calculus,  it  soon  permeates  all 
analysis,  and  toward  the  close  of  the  century  it  overhauls  and 
recasts  the  foundations  of  geometry  and  aspires  to  further 
conquests  in  mechanics  and  in  the  immense  domains  of  mathe- 
matical physics.  ...  A  searching  examination  of  the  founda- 
tions of  arithmetic  and  the  calculus  has  brought  to  light  the 
insufficiency  of  much  of  the  reasoning  formerly  considered  as 
conclusive. — PIERPONT,  J. 

History  of  Mathematics  in  the  Nineteenth 
Century;  Congress  of  Arts  and  Sciences  (Bos- 
ton and  New  York,  1905),  Vol.  1,  p.  482. 

718.  If  we  compare  a  mathematical  problem  with  an  immense 
rock,  whose  interior  we  wish  to  penetrate,  then  the  work  of  the 
Greek  mathematicians  appears  to  us  like  that  of  a  robust 
stonecutter,  who,  with  indefatigable  perseverance,  attempts  to 
demolish  the  rock  gradually  from  the  outside  by  means  of  ham- 
mer and  chisel;  but  the  modern  mathematician  resembles  an 
expert  miner,  who  first  constructs  a  few  passages  through  the 
rock  and  then  explodes  it  with  a  single  blast,  bringing  to  light  its 
inner  treasures. — HANKEL,  HERMANN. 

Die  Entwickelung  der  Mathematik  in  den 
letzten  Jahrhunderten  (Tubingen,  1884),  P-  9. 

719.  All  the  modern  higher  mathematics  is  based  on  a  calculus 
of  operations,  on  laws  of  thought.    All  mathematics,  from  the 
first,  was  so  in  reality;  but  the  evolvers  of  the  modern  higher 
calculus  have  known  that  it  is  so.    Therefore  elementary  teach- 
ers who,  at  the  present  day,  persist  in  thinking  about  algebra 
and  arithmetic  as  dealing  with  laws  of  number,  and  about 
geometry  as  dealing  with  laws  of  surface  and  solid  content,  are 
doing  the  best  that  in  them  lies  to  put  their  pupils  on  the  wrong 
track  for  reaching  in  the  future  any  true  understanding  of  the 
higher  algebras.    Algebras  deal  not  with  laws  of  number,  but 
with  such  laws  of  the  human  thinking  machinery  as  have  been 
discovered  in  the  course  of  investigations  on  numbers.    Plane 
geometry  deals  with  such  laws  of  thought  as  were  discovered  by 
men  intent  on  finding  out  how  to  measure  surface;  and  solid 
geometry  with  such  additional  laws  of  thought  as  were  dis- 


MODERN   MATHEMATICS  115 

covered  when  men  began  to  extend  geometry  into  three  dimen- 
sions.— BOOLE,  M.  E. 

Logic  of  Arithmetic  (Oxford,  1908),  Preface, 
pp.  18-19. 

720.  It  is  not  only  a  decided  preference  for  synthesis  and  a 
complete  denial  of  general  methods  which  characterizes  the 
ancient  mathematics  as  against  our  newer  science  [modern 
mathematics]:  besides  this  external  formal  difference  there  is 
another  real,  more  deeply  seated,  contrast,  which  arises  from  the 
different  attitudes  which  the  two  assumed  relative  to  the  use  of 
the  concept  of  variability.    For  while  the  ancients,  on  account  of 
considerations  which  had  been  transmitted  to  them  from  the 
philosophic  school  of  the  Eleatics,  never  employed  the  concept 
of  motion,  the  spatial  expression  for  variability,  in  their  rigor- 
ous system,  and  made  incidental  use  of  it  only  in  the  treatment 
of  phonoromically  generated  curves,  modern  geometry  dates 
from  the  instant  that  Descartes  left  the  purely  algebraic  treat- 
ment of  equations  and  proceeded  to  investigate  the  variations 
which  an  algebraic  expression  undergoes  when  one  of  its  variables 
assumes  a  continuous  succession  of  values. — HANKEL,  HERMANN. 

Untersuchungen  uber  die  unendlich  oft  oszil- 
lierenden  und  unstetigen  Functionen;  Ostwald's 
Klassiker  der  exacten  Wissenschaften,  No.  158, 
pp.  44-45- 

721.  Without  doubt  one  of  the  most  characteristic  features  of 
mathematics  in  the  last  century  is  the  systematic  and  universal 
use  of  the  complex  variable.    Most  of  its  great  theories  received 
invaluable  aid  from  it,  and  many  owe  their  very  existence  to  it. 

PlERPONT,    J. 

History  of  Mathematics  in  the  Nineteenth 
Century;  Congress  of  Arts  and  Sciences  (Bos- 
ton and  New  York,  1905),  Vol.  1,  p.  474. 

722.  The  notion,  which  is  really  the  fundamental  one  (and  I 
cannot  too  strongly  emphasise  the  assertion),  underlying  and 
pervading  the  whole  of  modern  analysis  and  geometry,  is  that  of 

maginary  magnitude  in  analysis  and  of  imaginary  space  in 
geometry. — CAYLEY,  ARTHUR. 

Presidential  Address;  Collected  Works,  Vol.  11, 

p.  434. 


116  MEMORABILIA   MATHEMATICA 

723.  The  solution  of  the  difficulties  which  formerly  surrounded 
the  mathematical  infinite  is  probably  the  greatest  achievement 
of  which  our  age  has  to  boast. — RUSSELL,  BERTRAND. 

The    Study    of    Mathematics;    Philosophical 
Essays  (London,  1910),  p.  77. 

724.  Induction  and  analogy  are  the  special  characteristics  of 
modern  mathematics,  in  which  theorems  have  given  place  to 
theories  and  no  truth  is  regarded  otherwise  than  as  a  link  in  an 
infinite  chain.    "Omne  exit  in  infinitum"  is  their  favorite  motto 
and  accepted  axiom. — SYLVESTER,  J.  J. 

A  Plea  for  the  Mathematician;  Nature,  Vol.  1 , 
p.  261. 

725.  The  conception  of  correspondence  plays  a  great  part  in 
modern  mathematics.     It  is  the  fundamental  notion  in  the 
science  of  order  as  distinguished  from  the  science  of  magnitude. 
If  the  older  mathematics  were  mostly  dominated  by  the  needs  of 
mensuration,  modern  mathematics  are  dominated  by  the  con- 
ception of  order  and  arrangement.    It  may  be  that  this  tendency 
of  thought  or  direction  of  reasoning  goes  hand  in  hand  with  the 
modern  discovery  in  physics,  that  the  changes  in  nature  depend 
not  only  or  not  so  much  on  the  quantity  of  mass  and  energy  as  on 
their  distribution  or  arrangement. — MERZ,  J.  T. 

History  of  European  Thought  in  the  Nineteenth 
Century  (Edinburgh  and  London,  1908),  p.  736. 

726.  Now  this  establishment  of  correspondence  between  two 
aggregates  and  investigation  of  the  propositions  that  are  carried 
over  by  the  correspondence  may  be  called  the  central  idea  of 
modern  mathematics. — CLIFFORD,  W.  K. 

Philosophy  of  the  Pure  Sciences;  Lectures  and 
Essays  (London,  1901),  Vol.  1,  p.  402. 

727.  In  our  century  the  conceptions  substitution  and  sub- 
stitution  group,    transformation    and    transformation    group, 
operation  and  operation  group,  invariant,  differential  invariant 
and  differential  parameter,  appear  more  and  more  clearly  as  the 
most  important  conceptions  of  mathematics. — LIE,  SOPHUS. 

Leipziger  Berichte,  No.  47  (1895),  p.  261. 


MODERN   MATHEMATICS  117 

728.  Generality  of  points  of  view  and  of  methods,  precision 
and  elegance  in  presentation,  have  become,  since  Lagrange,  the 
common  property  of  all  who  would  lay  claim  to  the  rank  of 
scientific  mathematicians.     And,  even  if  this  generality  leads 
at  times  to  abstruseness  at  the  expense  of  intuition  and  ap- 
plicability,   so   that  general  theorems  are  formulated  which 
fail  to  apply  to  a  single  special  case,  if  furthermore  precision 
at  times  degenerates  into  a  studied  brevity  which  makes  it  more 
difficult  to  read  an  article  than  it  was  to  write  it;  if,  finally, 
elegance  of  form  has  well-nigh  become  in  our  day  the  criterion  of 
the  worth  or  worthlessness  of  a  proposition, — yet  are  these 
conditions  of  the  highest  importance  to  a  wholesome  develop- 
ment, in  that  they  keep  the  scientific  material  within  the  limits 
which   are   necessary   both   intrinsically   and   extrinsically   if 
mathematics  is  not  to  spend  itself  in  trivialities  or  smother  in 
profusion. — HANKEL,  HERMANN. 

Die  Entwickelung  der  Mathematik  in  den  letzten 
Jahrhunderten  (Tubingen,  1884),  PP- 14~15. 

729.  The  development  of  abstract  methods  during  the  past 
few  years  has  given  mathematics  a  new  and  vital  principle  which 
furnishes  the  most  powerful   instrument   for  exhibiting  the 
essential  unity  of  all  its  branches. — YOUNG,  J.  W. 

Fundamental  Concepts  of  Algebra  and  Geomtry 
(New  York,  1911),  p.  225. 

730.  Everybody   praises    the    incomparable    power   of   the 
mathematical  method,  but  so  is  everybody  aware  of  its  incom- 
parable unpopularity. — ROSANES,  J. 

Jahresbericht    der    Deutschen    Mathematiker 
Vereinigung,  Bd.  18,  p.  17. 

731.  Indeed  the  modern  developments  of  mathematics  con- 
stitute not  only  one  of  the  most  impressive,  but  one  of  the  most 
characteristic,  phenomena  of  our  age.     It  is  a  phenomenon, 
however,  of  which  the  boasted  intelligence  of  a  "universalized" 
daily  press  seems  strangely  unaware;  and  there  is  no  other  great 
human  interest,  whether  of  science  or  of  art,  regarding  which 
the  mind  of  the  educated  public  is  permitted  to  hold  so  many 
fallacious    opinions   and   inferior    estimates. — KEYSER,   C.   J. 

Lectures  on  Science,  Philosophy  and  Arts  (New 
York,  1908),  p.  8. 


118  MEMORABILIA   MATHEMATICA 

732.  It   may   be   asserted   without   exaggeration   that   the 
domain  of  mathematical  knowledge  is  the  only  one  of  which  our 
otherwise  omniscient  journalism  has  not  yet  possessed  itself. 

PRINGSHEIM,  ALFRED. 

L/eoer  Wert  und  angeblichen  Unwert  der  Mathe- 
matik;  Jahresbericht  der  Deutschen  Mathema- 
tiker  Vereinigung,  (1904)  P-  357. 

733.  [The]  inaccessibility  of  special  fields  of  mathematics, 
except  by  the  regular  way  of  logically  antecedent  acquirements, 
renders  the  study  discouraging  or  hateful  to  weak  or  indolent 
minds. — LEFEVRE,  ARTHUR. 

Number    and    its    Algebra    (Boston,    1903), 


734.  The  majority  of  mathematical  truths  now  possessed  by 
us  presuppose  the  intellectual  toil  of  many  centuries.  A  mathe- 
matician, therefore,  who  wishes  today  to  acquire  a  thorough 
understanding  of  modern  research  in  this  department,  must 
think  over  again  in  quickened  tempo  the  mathematical  labors  of 
several  centuries.  This  constant  dependence  of  new  truths  on 
old  ones  stamps  mathematics  as  a  science  of  uncommon  ex- 
clusiveness  and  renders  it  generally  impossible  to  lay  open  to 
uninitiated  readers  a  speedy  path  to  the  apprehension  of  the 
higher  mathematical  truths.  For  this  reason,  too,  the  theories 
and  results  of  mathematics  are  rarely  adapted  for  popular 
presentation  .  .  .  This  same  inaccessibility  of  mathematics, 
although  it  secures  for  it  a  lofty  and  aristocratic  place  among  the 
sciences,  also  renders  it  odious  to  those  who  have  never  learned 
it,  and  who  dread  the  great  labor  involved  in  acquiring  an  under- 
standing of  the  questions  of  modern  mathematics.  Neither  in 
the  languages  nor  in  the  natural  sciences  are  the  investigations 
and  results  so  closely  interdependent  as  to  make  it  impossible 
to  acquaint  the  uninitiated  student  with  single  branches  or 
with  particular  results  of  these  sciences,  without  causing  him 
to  go  through  a  long  course  of  preliminary  study. 

SCHUBERT,  H. 

Mathematical  Essays  and  Recreations  (Chicago, 
1898),  p.  82. 


MODERN    MATHEMATICS  119 

735.  Such  is  the  character  of  mathematics  in  its  profounder 
depths  and  in  its  higher  and  remoter  zones  that  it  is  well  nigh 
impossible  to  convey  to  one  who  has  not  devoted  years  to  its 
exploration  a  just  impression  of  the  scope  and  magnitude  of  the 
existing  body  of  the  science.    An  imagination  formed  by  other 
disciplines  and  accustomed  to  the  interests  of  another  field  may 
scarcely  receive  suddenly  an  apocalyptic  vision  of  that  infinite 
interior  world.    But  how  amazing  and  how  edifying  were  such  a 
revelation,  if  it  only  could  be  made. — KEYSER,  C.  J. 

Lectures  on  Science,  Philosophy  and  Art  (New 
York,  1908),  p.  6. 

736.  It  is  not  so  long  since,  during  one  of  the  meetings  of  the 
Association,  one  of  the  leading  English  newspapers  briefly  de- 
scribed a  sitting  of  this  Section  in  the  words,  "Saturday  morn- 
ing was  devoted  to  pure  mathematics,  and  so  there  was  nothing 
of  any  general  interest:"  still,  such  toleration  is  better  than 
undisguised  and  ill-informed  hostility. — FORSYTH,  A.  R. 

Report  of  the  67th  meeting  of  the  British 
Association  for  the  Advancement  of  Science. 

737.  The  science  [of  mathematics]  has  grown  to  such  vast 
proportion  that  probably  no  living  mathematician  can  claim  to 
have  achieved  its  mastery  as  a  whole. — WHITEHEAD,  A.  N. 

An  Introduction  to  Mathematics  (New  York, 
1911),  p.  252. 

738.  There  is  perhaps  no  science  of  which  the  development 
has  been  carried  so  far,  which  requires  greater  concentration 
and  will  power,  and  which  by  the  abstract  height  of  the  qualities 
required  tends  more  to  separate  one  from  daily  life. 

Provisional  Report  of  the  American  Subcom- 
mittee of  the  International  Commission  on  the 
Teaching  of  Mathematics;  Bulletin  American 
Society  (1910),  p.  97. 

739.  Angling  may  be  said  to  be  so  like  the  mathematics,  that 
it  can  never  be  fully  learnt. — WALTON,  ISAAC. 

The  Complete  Angler,  Preface. 


120  MEMORABILIA   MATHEMATICA 

740.  The  flights  of  the  imagination  which  occur  to  the  pure 
mathematician  are  in  general  so  much  better  described  in  his 
formulae  than  in  words,  that  it  is  not  remarkable  to  find  the 
subject  treated  by  outsiders  as  something  essentially  cold  and 
uninteresting —  .  .  .  the  only  successful  attempt  to  invest 
mathematical  reasoning  with  a  halo  of  glory — that  made  in  this 
section  by  Prof.  Sylvester — is  known  to  a  comparative  few,  .  .  . 

TAIT,  P.  G. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science  (1871);  Nature 
Vol.  4,  P.  ML 


CHAPTER  VIII 

THE   MATHEMATICIAN 

801.  The  real  mathematician  is  an  enthusiast  per  se.    With- 
out enthusiasm  no  mathematics. — NOVALIS. 

Schriften  (Berlin,  1901),  Zweiter  Teil,  p.  223. 

802.  It  is  true  that  a  mathematician,  who  is  not  somewhat  of 
a  poet,  will  never  be  a  perfect  mathematician. — WEIERSTRASS. 

Quoted  by  Mittag-Leffler;  Compte  rendu  du 
deuxieme  congres  international  des  mathema- 
ticiens  (Paris,  1902},  p.  149. 

803.  The  mathematician  is  perfect  only  in  so  far  as  he  is  a 
perfect  being,  in  so  far  as  he  perceives  the  beauty  of  truth;  only 
then  will  his  work  be  thorough,  transparent,  comprehensive, 
pure,  clear,  attractive  and  even  elegant.    All  this  is  necessary  to 
resemble  Lagrange. — GOETHE. 

Wilhelm  Meister's  Wanderjahre,  Zweites 
Buck;  Spriiche  in  Prosa;  Natur,  VI,  950. 

804.  A  thorough  advocate  in  a  just  cause,  a  penetrating 
mathematician  facing  the  starry  heavens,  both  alike  bear  the 
semblance  of  divinity. — GOETHE. 

Wilhelm  Meister's  Wanderjahre,  Zweites  Buck. 
Spruche  in  Prosa;  Natur,  VI,  947. 

805.  Mathematicians  practice  absolute  freedom. 

ADAMS,  HENRY. 

A  Letter  to  American  Teachers  of  History 
(Washington,  1910),  p.  169. 

806.  The  mathematical  method  is  the  essence  of  mathematics. 
He  who  fully  comprehends  the  method  is  a  mathematician. 

NOVALIS. 

Schriften  (Berlin,  1901),  Zweiter  Teil,  p.  190. 
121 


122  MEMORABILIA   MATHEMATICA 

807.  He  who  is  unfamiliar  with  mathematics  [literally,  he 
who  is  a  layman  in  mathematics]  remains  more  or  less  a  stranger 
to  our  time. — DILLMANN,  E. 

Die  Mathematik  die  Fackeltrdgerin  einer  neuen 
Zeit  (Stuttgart,  1889),  p.  39. 

808.  Enlist    a   great    mathematician   and    a    distinguished 
Grecian;  your  problem  will  be  solved.    Such  men  can  teach  in  a 
dwelling-house  as  well  as  in  a  palace.    Part  of  the  apparatus 
they  will  bring;  part  we  will  furnish.     [Advice  given  to  the 
Trustees  of  Johns  Hopkins  University  on  the  choice  of  a  pro- 
fessorial staff.] — OILMAN,  D.  C. 

Report  of  the  President  of  Johns  Hopkins 
University  (1888),  p.  29. 

809.  Persons,  who  have  a  decided  mathematical  talent,  con- 
stitute, as  it  were,  a  favored  class.    They  bear  the  same  relation 
to  the  rest  of  mankind  that  those  who  are  academically  trained 
bear  to  those  who  are  not. — MOEBIUS,  P.  J. 

Ueber  die  Anlage  zur  Mathematik  (Leipzig, 
1900),  p.  4- 

x 

810.  One  may  be  a  mathematician  of  the  first  rank  without 
being  able  to  compute.    It  is  possible  to  be  a  great  computer 
without  having  the  slightest  idea  of  mathematics. — NOVALIS. 

Schriften,  Zweiter  Teil  (Berlin,  1901),  p.  223. 

811.  It  has  long  been  a  complaint  against  mathematicians 
that  they  are  hard  to  convince :  but  it  is  a  far  greater  disqualifi- 
cation both  for  philosophy,  and  for  the  affairs  of  life,  to  be  too 
easily  convinced;  to  have  too  low  a  standard  of  proof.    The  only 
sound  intellects  are  those  which,  in  the  first  instance,  set  their 
standards  of  proof  high.     Practice  in  concrete  affairs  soon 
teaches  them  to  make  the  necessary  abatement :  but  they  retain 
the  consciousness,  without  which  there  is  no  sound  practical 
reasoning,  that  in  accepting  inferior  evidence  because  there  is 
no  better  to  be  had,  they  do  not  by  that  acceptance  raise  it  to 
completeness. — MILL,  J.  S. 

An  Examination  of  Sir  William  Hamilton's 
Philosophy  (London,  1878),  p.  611. 


THE   MATHEMATICIAN  123 

812.  It  is  easier  to  square  the  circle  than  to  get  round  a 
mathematician. — DE  MORGAN,  A. 

Budget  of  Paradoxes  (London,  1873),  p.  90. 

813.  Mathematicians  are  like  Frenchmen:  whatever  you  say 
to  them  they  translate  into  their  own  language  and  forthwith  it 
is  something  entirely  different. — GOETHE. 

Maximen  und  Reflexionen,  Sechste  Abtheilung. 

814.  What  I  chiefly  admired,  and  thought  altogether  un- 
accountable, was  the  strong  disposition  I  observed  in  them 
[the  mathematicians  of  Laputa]  towards  news  and  politics; 
perpetually  inquiring  into  public  affairs;  giving  their  judgments 
in  matters  of  state;  and  passionately  disputing  every  inch  of 
party  opinion.     I  have  indeed  observed  the  same  disposition 
among  most  of  the  mathematicians  I  have  known  in  Europe, 
although  I  could  never  discover  the  least  analogy  between  the 
two  sciences. — SWIFT,  JONATHAN. 

Gulliver's  Travels,  Part  3,  chap.  2. 

816.  The  great  mathematician,  like  the  great  poet  or  natural- 
ist or  great  administrator,  is  born.  My  contention  shall  be  that 
where  the  mathematic  endowment  is  found,  there  will  usually 
be  found  associated  with  it,  as  essential  implications  in  it,  other 
endowments  in  generous  measure,  and  that  the  appeal  of  the 
science  is  to  the  whole  mind,  direct  no  doubt  to  the  central 
powers  of  thought,  but  indirectly  through  sympathy  of  all, 
rousing,  enlarging,  developing,  emancipating  all,  so  that  the 
faculties  of  will,  of  intellect  and  feeling  learn  to  respond,  each 
in  its  appropriate  order  and  degree,  like  the  parts  of  an  orchestra 
to  the  "urge  and  ardor"  of  its  leader  and  lord. — KEYSER,  C.  J. 

Lectures  on  Science,  Philosophy  and  Art  (New 

York,  1908),  p.  22. 

816.  Whoever  limits  his  exertions  to  the  gratification  of 
others,  whether  by  personal  exhibition,  as  in  the  case  of  the  actor 
and  of  the  mimic,  or  by  those  kinds  of  literary  composition 
which  are  calculated  for  no  end  but  to  please  or  to  entertain, 
renders  himself,  in  some  measure,  dependent  on  their  caprices 
and  humours.  The  diversity  among  men,  in  their  judgments 


124  MEMORABILIA   MATHEMATICA 

concerning  the  objects  of  taste,  is  incomparably  greater  than  in 
their  speculative  conclusions;  and  accordingly,  a  mathematician 
will  publish  to  the  world  a  geometrical  demonstration,  or  a 
philosopher,  a  process  of  abstract  reasoning,  with  a  confidence 
very  different  from  what  a  poet  would  feel,  in  communicating 
one  of  his  productions  even  to  a  friend. — STEWART,  DUGALD. 

Elements  of  the  Philosophy  of  the  Human 
Mind,  Part  8,  chap.  1,  sect.  3. 

817.  Considering  that,  among  all  those  who  up  to  this  time 
made  discoveries  in  the  sciences,  it  was  the  mathematicians 
alone  who  had  been  able  to  arrive  at  demonstrations — that  is  to 
say,  at  proofs  certain  and  evident — I  did  not  doubt  that  I 
should  begin  with  the  same  truths  that  they  have  investigated, 
although  I  had  looked  for  no  other  advantage  from  them  than  to 
accustom  my  mind  to  nourish  itself  upon  truths  and  not  to  be 
satisfied  with  false  reasons. — DESCARTES. 

Discourse  upon  Method,  Part  2;    Philosophy 
of  Descartes  [Torrey]  (New  York,  1892},  p.  48. 

818.  When  the  late  Sophus  Lie  .  .  .  was  asked  to  name  the 
characteristic  endowment  of  the  mathematician,  his  answer  was 
the  following  quaternion:  Phantasie,  Energie,  Selbstvertrauen, 
Selbstkritik. — KEYSER,  C.  J. 

Lectures  on  Philosophy,  Science  and  Art  (New 
York,  1908),  p.  31. 

819.  The  existence  of  an  extensive  Science  of  Mathematics, 
requiring  the  highest  scientific  genius  in  those  who  contributed 
to  its  creation,  and  calling  for  the  most  continued  and  vigorous 
exertion  of  intellect  in  order  to  appreciate  it  when  created,  etc. 

MILL,  J.  S. 
System  of  Logic,  Bk.  2,  chap.  4,  sect.  4- 

820.  It  may  be  true,  that  men,  who  are  mere  mathematicians, 
have  certain  specific  shortcomings,  but  that  is  not  the  fault  of 
mathematics,  for  it  is  equally  true  of  every  other  exclusive 
occupation.    So  there  are  mere  philologists,  mere  jurists,  mere 
soldiers,  mere  merchants,  etc.    To  such  idle  talk  it  might  further 
be  added:  that  whenever  a  certain  exclusive  occupation  is 


THE   MATHEMATICIAN  125 

coupled  with  specific  shortcomings,  it  is  likewise  almost  certainly 
divorced  from  certain  other  shortcomings. — GAUSS. 

Gauss-Schumacher  Brief wechsel,  Bd.  4, 

(Altona,  1862],  p.  387. 

821.  Mathematical    studies  .  .  .  when   combined,    as   they 
now  generally  are,  with  a  taste  for  physical  science,  enlarge 
infinitely  our  views  of  the  wisdom  and  power  displayed  in  the 
universe.    The  very  intimate  connexion  indeed,  which,  since  the 
date  of  the  Newtonian  philosophy,  has  existed  between  the 
different  branches  of  mathematical  and  physical  knowledge, 
renders  such  a  character  as  that  of  a  mere  mathematician  a  very 
rare   and   scarcely   possible   occurrence. — STEWART,   DUGALD. 

Elements  of  the  Philosophy  of  the  Human 
Mind,  part  3,  chap.  1,  sect.  3. 

822.  Once  when  lecturing  to  a  class  he  [Lord  Kelvin]  used  the 
word  "mathematician,"  and  then  interrupting  himself  asked  his 
class:  "Do  you  know  what  a  mathematician  is?"    Stepping  to 
the  blackboard  he  wrote  upon  it : — 

/«• 

e  Ax~    AT 

oo 

Then  putting  his  finger  on  what  he  had  written,  he  turned  to 
his  class  and  said :  "  A  mathematician  is  one  to  whom  that  is  as 
obvious  as  that  twice  two  makes  four  is  to  you.  Liouville  was  a 
mathematician — THOMPSON,  S.  P. 

Life  of  Lord  Kelvin  (London,  1910),  p.  1139. 

823.  It  is  not  surprising,  in  view  of  the  polydynamic  con- 
stitution of  the  genuinely  mathematical  mind,  that  many  of  the 
major  heros  of  the  science,  men  like  Desargues  and  Pascal, 
Descartes  and  Leibnitz,  Newton,  Gauss  and  Bolzano,  Helm- 
holtz  and  Clifford,  Riemann  and  Salmon  and  Pliicker  and 
Poincare,  have  attained  to  high  distinction  in  other  fields  not 
only  of  science  but  of  philosophy  and  letters  too.    And  when 
we  reflect  that  the  very  greatest  mathematical  achievements 
have  been  due,  not  alone  to  the  peering,  microscopic,  histologic 
vision  of  men  like  Weierstrass,  illuminating  the  hidden  recesses, 


126  MEMORABILIA   MATHEMATICA 

the  minute  and  intimate  structure  of  logical  reality,  but  to  the 
larger  vision  also  of  men  like  Klein  who  survey  the  kingdoms  of 
geometry  and  analysis  for  the  endless  variety  of  things  that 
flourish  there,  as  the  eye  of  Darwin  ranged  over  the  flora  and 
fauna  of  the  world,  or  as  a  commercial  monarch  contemplates 
its  industry,  or  as  a  statesman  beholds  an  empire;  when  we 
reflect  not  only  that  the  Calculus  of  Probability  is  a  creation 
of  mathematics  but  that  the  master  mathematician  is  constantly 
required  to  exercise  judgment — judgment,  that  is,  in  matters 
not  admitting  of  certainty — balancing  probabilities  not  yet 
reduced  nor  even  reducible  perhaps  to  calculation;  when  we 
reflect  that  he  is  called  upon  to  exercise  a  function  analogous 
to  that  of  the  comparative  anatomist  like  Cuvier,  comparing 
theories  and  doctrines  of  every  degree  of  similarity  and  dis- 
similarity of  structure;  when,  finally,  we  reflect  that  he  seldom 
deals  with  a  single  idea  at  a  time,  but  is  for  the  most  part  en- 
gaged in  wielding  organized  hosts  of  them,  as  a  general  wields  at 
once  the  division  of  an  army  or  as  a  great  civil  administrator 
directs  from  his  central  office  diverse  and  scattered  but  related 
groups  of  interests  and  operations;  then,  I  say,  the  current 
opinion  that  devotion  to  mathematics  unfits  the  devotee  for 
practical  affairs  should  be  known  for  false  on  a  priori  grounds. 
And  one  should  be  thus  prepared  to  find  that  as  a  fact  Gaspard 
Monge,  creator  of  descriptive  geometry,  author  of  the  classic 
"Applications  de  1 'analyse  a  la  ge'ome'trie";  Lazare  Carnot, 
author  of  the  celebrated  works,  "Ge'ome'trie  de  position,"  and 
"Reflections  sur  la  Me"taphysique  du  Calcul  infinitesimal"; 
Fourier,  immortal  creator  of  the  "The"orie  analytique  de  la 
chaleur";  Arago,  rightful  inheritor  of  Monge's  chair  of  geome- 
try; Poncelet,  creator  of  pure  protective  geometry;  one  should 
not  be  surprised,  I  say,  to  find  that  these  and  other  mathe- 
maticians in  a  land  sagacious  enough  to  invoke  their  aid,  ren- 
dered, alike  in  peace  and  in  war,  eminent  public  service. 

KEYSER,  C.  J. 

Lectures  on  Science,  Philosophy  and  Art  (New 
York,  1908),  pp.  32-38. 

824.  If  in  Germany  the  goddess  Justitia  had  not  the  unfortu- 
nate habit  of  depositing  the  ministerial  portfolios  only  in  the 


THE   MATHEMATICIAN  127 

cradles  of  her  own  progeny,  who  knows  how  many  a  German 
mathematician  might  not  also  have  made  an  excellent  minister. 

PRINGSHEIM,  A. 

Jahresbericht    der    Deutschen    Mathematiker 
Vereinigung,  Bd.  13  (1904),  P>  372. 

825.  We  pass  with  admiration  along  the  great  series  of  mathe- 
maticians, by  whom  the  science  of  theoretical  mechanics  has 
been  cultivated,  from  the  time  of  Newton  to  our  own.    There  is 
no  group  of  men  of  science  whose  fame  is  higher  or  brighter. 
The  great  discoveries  of  Copernicus,  Galileo,  Newton,  had  fixed 
all  eyes  on  those  portions  of  human  knowledge  on  which  their 
successors  employed  their  labors.    The  certainty  belonging  to 
this  line  of  speculation  seemed  to  elevate  mathematicians  above 
the  students  of  other  subjects;  and  the  beauty  of  mathematical 
relations  and  the  subtlety  of  intellect  which  may  be  shown  in 
dealing  with  them,  were  fitted  to  win  unbounded  applause. 
The  successors  of  Newton  and  the  Bernoullis,  as  Euler,  Clairaut, 
D'Alembert,  Lagrange,  Laplace,  not  to  introduce  living  names, 
have  been  some  of  the  most  remarkable  men  of  talent  which  the 
world  has  seen. — WHEWELL.  W. 

History  of  the  Inductive  Sciences,  Vol.  1,  Bk.  4, 
chap.  6,  sect.  6. 

826.  The  persons  who  have  been  employed  on  these  problems 
of  applying  the  properties  of  matter  and  the  laws  of  motion  to 
the  explanation  of  the  phenomena  of  the  world,  and  who  have 
brought  to  them  the  high  and  admirable  qualities  which  such  an 
office  requires,  have  justly  excited  in  a  very  eminent  degree  the 
admiration  which  mankind  feels  for  great  intellectual  powers. 
Their  names  occupy  a  distinguished  place  in  literary  history;  and 
probably  there  are  no  scientific  reputations  of  the  last  century 
higher,  and  none  more  merited,  than  those  earned  by  great 
mathematicians  who  have  laboured  with  such  wonderful  success 
in  unfolding  the  mechanism  of  the  heavens;  such  for  instance  as 
D'Alembert,  Clairaut,  Euler,  Lagrange,  Laplace. 

WHEWELL,  W. 

Astronomy    and    General    Physics    (London, 
1888),  Bk.  8,  chap.  4,  p.  827. 


128  MEMORABILIA    MATHEMATICA 

827.  Two  extreme  views  have  always  been  held  as  to  the  use  of 
mathematics.  To  some,  mathematics  is  only  measuring  and 
calculating  instruments,  and  their  interest  ceases  as  soon  as 
discussions  arise  which  cannot  benefit  those  who  use  the  in- 
struments for  the  purposes  of  application  in  mechanics,  astron- 
omy, physics,  statistics,  and  other  sciences.  At  the  other  ex- 
treme we  have  those  who  are  animated  exclusively  by  the  love  of 
pure  science.  To  them  pure  mathematics,  with  the  theory  of 
numbers  at  the  head,  is  the  only  real  and  genuine  science,  and 
the  applications  have  only  an  interest  in  so  far  as  they  contain 
or  suggest  problems  in  pure  mathematics. 

Of  the  two  greatest  mathematicians  of  modern  times,  Newton 
and  Gauss,  the  former  can  be  considered  as  a  representative  of 
the  first,  the  latter  of  the  second  class;  neither  of  them  was 
exclusively  so,  and  Newton's  inventions  in  the  science  of  pure 
mathematics  were  probably  equal  to  Gauss's  work  in  applied 
mathematics.  Newton's  reluctance  to  publish  the  method  of 
fluxions  invented  and  used  by  him  may  perhaps  be  attributed  to 
the  fact  that  he  was  not  satisfied  with  the  logical  foundations 
of  the  Calculus;  and  Gauss  is  known  to  have  abandoned  his 
electro-dynamic  speculations,  as  he  could  not  find  a  satisfying 
physical  basis.  .  .  . 

Newton's  greatest  work,  the  "  Principia  ",  laid  the  foundation 
of  mathematical  physics;  Gauss's  greatest  work,  the  "  Disquisi- 
tiones  Arithmeticae  ",  that  of  higher  arithmetic  as  distinguished 
from  algebra.  Both  works,  written  in  the  synthetic  style  of  the 
ancients,  are  difficult,  if  not  deterrent,  in  their  form,  neither  of 
them  leading  the  reader  by  easy  steps  to  the  results.  It  took 
twenty  or  more  years  before  either  of  these  works  received  due 
recognition;  neither  found  favour  at  once  before  that  great 
tribunal  of  mathematical  thought,  the  Paris  Academy  of 
Sciences.  .  .  . 

The  country  of  Newton  is  still  pre-eminent  for  its  culture  of 
mathematical  physics,  that  of  Gauss  for  the  most  abstract 
work  in  mathematics. — MERZ,  J.  T. 

History  of  European  Thought  in  the  Nine- 
teenth Century  (Edinburgh  and  London,  1903), 
p.  630. 


THE   MATHEMATICIAN  129 

828.  As  there  is  no  study  which  may  be  so  advantageously 
entered  upon  with  a  less  stock  of  preparatory  knowledge  than 
mathematics,  so  there  is  none  in  which  a  greater  number  of 
uneducated  men  have  raised  themselves,  by  their  own  exertions, 
to  distinction  and  eminence.  .  .  .    Many  of  the  intellectual  de- 
fects which,  in  such  cases,  are  commonly  placed  to  the  account 
of  mathematical  studies,  ought  to  be  ascribed  to  the  want  of  a 
liberal  education  in  early  youth. — STEWART,  DUGALD. 

Elements  of  the  Philosophy  of  the  Human 
Mind,  Part  3,  chap.  1,  sect.  8. 

829.  I  know,  indeed,  and  can  conceive  of  no  pursuit  so  antag- 
onistic to  the  cultivation  of  the  oratorical  faculty  ...  as  the 
study  of  Mathematics.    An  eloquent  mathematician  must,  from 
the  nature  of  things,  ever  remain  as  rare  a  phenomenon  as  a 
talking  fish,  and  it  is  certain  that  the  more  anyone  gives  himself 
up  to  the  study  of  oratorical  effect  the  less  will  he  find  himself  in 
a  fit  state  to  mathematicize.     It  is  the  constant  aim  of  the 
mathematician  to  reduce  all  his  expressions  to  their  lowest 
terms,  to  retrench  every  superfluous  word  and  phrase,  and  to 
condense  the  Maximum  of  meaning  into  the  Minimum  of  lan- 
guage.   He  has  to  turn  his  eye  ever  inwards,  to  see  everything  in 
its  dryest  light,  to  train  and  inure  himself  to  a  habit  of  internal 
and  impersonal  reflection  and  elaboration  of  abstract  thought, 
which  makes  it  most  difficult  for  him  to  touch  or  enlarge  upon 
any  of  those  themes  which  appeal  to  the  emotional  nature  of 
his  fellow-men.    When  called  upon  to  speak  in  public  he  feels 
as  a  man  might  do  who  has  passed  all  his  life  in  peering  through 
a  microscope,  and  is  suddenly  called  upon  to  take  charge  of  a 
astronomical  observatory.    He  has  to  get  out  of  himself,  as  it 
were,  and  change  the  habitual  focus  of  his  vision. 

SYLVESTER,  J.  J. 

Baltimore    Address;    Mathematical    Papers, 
Vol.  S,  pp.  72-78. 

830.  An  accomplished  mathematician,  i.  e.  a  most  wretched 
orator. — BARROW,  ISAAC. 

Mathematical    Lectures     (London,    1734), 
p.  32. 


130  MEMORABILIA   MATHEMATICA 

831.  Nemo    mathematicus  genium  indemnatus   habebit.    [No 
mathematician  *  is  esteemed  a  genius  until  condemned.] 

Juvenal,  Liberii,  Satura  VI,  562. 

832.  Taking  .  .  .  the  mathematical  faculty,  probably  fewer 
than  one  in  a  hundred  really  possess  it,  the  great  bulk  of  the 
population  having  no  natural  ability  for  the  study,  or  feeling  the 
slightest  interest  in  it.f    And  if  we  attempt  to  measure  the 
amount  of  variation  in  the  faculty  itself  between  a  first-class 
mathematician  and  the  ordinary  run  of  people  who  find  any 
kind  of  calculation  confusing  and  altogether  devoid  of  interest, 
it  is  probable  that  the  former  could  not  be  estimated  at  less  than 
a  hundred  times  the  latter,  and  perhaps  a  thousand  times  would 
more  nearly  measure  the  difference  between  them. 

WALLACE,  A.  R. 
Darwinism,  chap.  15. 

833.  .  .  .  the  present  gigantic  development  of  the  mathe- 
matical faculty  is  wholly  unexplained  by  the  theory  of  natural 
selection,  and  must  be  due  to  some  altogether  distinct  cause. 

WALLACE,  A.  R. 
Darwinism,  chap.  15. 

834.  Dr.  Wallace,  in  his  "  Darwinism  ",  declares  that  he  can 
find  no  ground  for  the  existence  of  pure  scientists,  especially 
mathematicians,  on  the  hypothesis  of  natural  selection.    If  we 
put  aside  the  fact  that  great  power  in  theoretical  science  is 
correlated  with  other  developments  of  increasing  brain-activity, 
we  may,  I  think,  still  account  for  the  existence  of  pure  scientists 
as  Dr.  Wallace  would  himself  account  for  that  of  worker-bees. 
Their  function  may  not  fit  them  individually  to  survive  in  the 
struggle  for  existence,  but  they  are  a  source  of  strength  and 
efficiency  to  the  society  which  produces  them. — PEARSON,  KARL. 

Grammar  of  Science  (London,  1911),  Part  1, 
p.  221. 

*  Used  here  in  the  sense  of  astrologer,  or  soothsayer. 

t  This  is  the  estimate  furnished  me  by  two  mathematical  masters  in 
one  of  our  great  public  schools  of  the  proportion  of  boys  who  have  any 
special  taste  or  capacity  for  mathematical  studies.  Many  more,  of 
course,  can  be  drilled  into  a  fair  knowledge  of  elementary  mathe- 
matics, but  only  this  small  proportion  possess  the  natural  faculty  which 
renders  it  possible  for  them  ever  to  rank  high  as  mathematicians,  to 
take  any  pleasure  in  it,  or  to  do  any  original  mathematical  work. 


THE   MATHEMATICIAN  131 

835.  It  is  only  in  mathematics,  and  to  some  extent  in  poetry, 
that  originality  may  be  attained  at  an  early  age,  but  even  then 
it  is  very  rare  (Newton  and  Keats  are  examples),  and  it  is  not 
notable  until   adolescence  is  completed. — ELLIS,   HAVELOCK. 

A  Study  of  British  Genius  (London,  1904), 
p.  142. 

836.  The  Anglo-Dane  appears  to  possess  an  aptitude  for 
mathematics  which  is  not  shared  by  the  native  of  any  other 
English  district  as  a  whole,  and  it  is  in  the  exact  sciences  that 
the  Anglo-Dane  triumphs.* — ELLIS,  HAVELOCK. 

A  Study  of  British  Genius  (London,  1904), 
p.  69. 

837.  In  the  whole  history  of  the  world  there  was  never  a 
race  with  less  liking  for  abstract  reasoning  than  the  Anglo- 
Saxon.  .  .  .    Common-sense  and  compromise  are  believed  in, 
logical  deductions  from  philosophical  principles  are  looked  upon 
with  suspicion,  not  only  by  legislators,  but  by  all  our  most 
learned  professional  men. — PERRY,  JOHN. 

The  Teaching  of  Mathematics  (London,  1902), 
pp.  20-21. 

838.  The  degree  of  exactness  of  the  intuition  of  space  may  be 
different  in  different  individuals,  perhaps  even  in  different  races. 
It  would  seem  as  if  a  strong  naive  space-intuition  were  an  at- 
tribute pre-eminently  of  the  Teutonic  race,  while  the  critical, 
purely  logical  sense  is  more  fully  developed  in  the  Latin  and 
Hebrew  races.    A  full  investigation  of  this  subject,  somewhat  on 
the  lines  suggested  by  Francis  Gallon  in  his  researches  on  hered- 
ity, might  be  interesting. — KLEIN,  FELIX. 

The  Evanston  Colloquium  Lectures  (New  York, 
1894),  p.  46. 

839.  This  [the  fact  that  the  pursuit  of  mathematics  brings 
into  harmonious  action  all  the  faculties  of  the  human  mind] 
accounts  for  the  extraordinary  longevity  of  all  the  greatest 
masters  of  the  Analytic  art,  the  Dii  Majores  of  the  mathematical 

*  The  mathematical  tendencies  of  Cambridge  are  due  to  the  fact 
that  Cambridge  drains  the  ability  of  nearly  the  whole  Anglo-Danish 
district. 


132  MEMORABILIA    MATHEMATICA 

Pantheon.  Leibnitz  lived  to  the  age  of  70;  Euler  to  76;  La- 
grange  to  77;  Laplace  to  78;  Gauss  to  78;  Plato,  the  supposed 
inventor  of  the  conic  sections,  who  made  mathematics  his 
study  and  delight,  who  called  them  the  handles  or  aids  to 
philosophy,  the  medicine  of  the  soul,  and  is  said  never  to  have 
let  a  day  go  by  without  inventing  some  new  theorems,  lived  to 
82;  Newton,  the  crown  and  glory  of  his  race,  to  85;  Archimedes, 
the  nearest  akin,  probably,  to  Newton  in  genius,  was  75,  and 
might  have  lived  on  to  be  100,  for  aught  we  can  guess  to  the 
contrary,  when  he  was  slain  by  the  impatient  and  ill-mannered 
sergeant,  sent  to  bring  him  before  the  Roman  general,  in  the  full 
vigour  of  his  faculties,  and  in  the  very  act  of  working  out  a 
problem;  Pythagoras,  in  whose  school,  I  believe,  the  word  math- 
ematician (used,  however,  in  a  somewhat  wider  than  its  present 
sense)  originated,  the  second  founder  of  geometry,  the  inventor 
of  the  matchless  theorem  which  goes  by  his  name,  the  pre- 
cognizer  of  the  undoubtedly  mis-called  Copernican  theory,  the 
discoverer  of  the  regular  solids  and  the  musical  canon  who 
stands  at  the  very  apex  of  this  pyramid  of  fame,  (if  we  may 
credit  the  tradition)  after  spending  22  years  studying  in  Egypt, 
and  12  in  Babylon,  opened  school  when  56  or  57  years  old  in 
Magna  Graecia,  married  a  young  wife  when  past  60,  and  died, 
carrying  on  his  work  with  energy  unspent  to  the  last,  at  the  age 
of  99.  The  mathematician  lives  long  and  lives  young;  the  wings 
of  his  soul  do  not  early  drop  off,  nor  do  its  pores  become  clogged 
with  the  earthy  particles  blown  from  the  dusty  highways  of 
vulgar  life. — SYLVESTER,  J.  J. 

Presidential  Address  to  the  British  Association; 

Collected  Mathematical  Papers,  Vol.  2  (1908), 

p.  658. 

840.  The  game  of  chess  has  always  fascinated  mathemati- 
cians, and  there  is  reason  to  suppose  that  the  possession  of 
great  powers  of  playing  that  game  is  in  many  features  very  much 
like  the  possession  of  great  mathematical  ability.  There  are  the 
different  pieces  to  learn,  the  pawns,  the  knights,  the  bishops, 
the  castles,  and  the  queen  and  king.  The  board  possesses  certain 
possible  combinations  of  squares,  as  in  rows,  diagonals,  etc. 
The  pieces  are  subject  to  certain  rules  by  which  their  motions  are 
governed,  and  there  are  other  rules  governing  the  players.  .  .  . 


THE   MATHEMATICIAN  133 

One  has  only  to  increase  the  number  of  pieces,  to  enlarge  the 
field  of  the  board,  and  to  produce  new  rules  which  are  to  govern 
either  the  pieces  or  the  player,  to  have  a  pretty  good  idea  of 
what  mathematics  consists. — SHAW,  J.  B. 

What  is  Mathematics?  Bulletin  American 
Mathematical  Society  Vol.  18  (1912),  pp.  886- 
387. 

841.  Every  man  is  ready  to  join  in  the  approval  or  condemna- 
tion of  a  philosopher  or  a  statesman,  a  poet  or  an  orator,  an 
artist  or  an  architect.    But  who  can  judge  of  a  mathematician? 
Who  will  write  a  review  of  Hamilton's  Quaternions,  and  show 
us  wherein  it  is  superior  to  Newton's  Fluxions? — HILL,  THOMAS. 

Imagination  in  Mathematics;  North  American 
Review,  Vol.  85,  p.  224. 

842.  The  pursuit  of  mathematical  science  makes  its  votary 
appear  singularly  indifferent  to  the  ordinary  interests  and  cares 
of  men.    Seeking  eternal  truths,  and  finding  his  pleasures  in  the 
realities  of  form  and  number,  he  has  little  interest  in  the  dis- 
putes and  contentions  of  the  passing  hour.    His  views  on  social 
and  political  questions  partake  of  the  grandeur  of  his  favorite 
contemplations,  and,  while  careful  to  throw  his  mite  of  influence 
on  the  side  of  right  and  truth,  he  is  content  to  abide  the  work- 
ings of  those  general  laws  by  which  he  doubts  not  that  the 
fluctuations  of  human  history  are  as  unerringly  guided  as  are  the 
perturbations  of  the  planetary  hosts. — HILL,  THOMAS. 

Imagination  in  Mathematics;  North  American 
Review,  Vol.  85,  p.  227. 

843.  There  is  something  sublime  in  the  secrecy  in  which  the 
really  great  deeds  of  the  mathematician  are  done.    No  popular 
applause  follows  the  act;  neither  contemporary  nor  succeeding 
generations  of  the  people  understand  it.    The  geometer  must 
be  tried  by  his  peers,  and  those  who  truly  deserve  the  title  of 
geometer  or  analyst  have  usually  been  unable  to  find  so  many 
as  twelve  living  peers  to  form  a  jury.    Archimedes  so  far  out- 
stripped his  competitors  in  the  race,  that  more  than  a  thousand 
years  elapsed  before  any  man  appeared,  able  to  sit  in  judgment 
on  his  work,  and  to  say  how  far  he  had  really  gone.    And  in 
judging  of  those  men-  whose  names  are  worthy  of  being  men- 


134  MEMORABILIA   MATHEMATICA 

tioned  in  connection  with  his, — Galileo,  Descartes,  Leibnitz, 
Newton,  and  the  mathematicians  created  by  Leibnitz  and 
Newton's  calculus, — we  are  forced  to  depend  upon  their  testi- 
mony of  one  another.  They  are  too  far  above  our  reach  for  us  to 
judge  of  them. — HILL,  THOMAS. 

Imagination  in  Mathematics;  North  American 

Review,  Vol.  85,  p.  223. 

844.  To  think  the  thinkable — that  is  the  mathematician's 
aim. — KEYSER,  C.  J. 

The  Universe  and  Beyond;  Hibbert  Journal, 
Vol.  3  (1904-1905),  p.  812. 

845.  Every  common  mechanic  has  something  to  say  in  his 
craft  about  good  and  evil,  useful  and  useless,  but  these  practical 
considerations  never  enter  into  the  purview  of  the  mathemati- 
cian.— ARISTIPPUS  THE  CYRE..AIC. 

Quoted  in  Hicks,  R.  D.,  Stoic  and  Epicurean, 
(New  York,  1910)  p.  210. 


CHAPTER  IX 

PERSONS   AND   ANECDOTES 

(A-M) 

901.  Alexander  is  said  to  have  asked  Menaechmus  to  teach 
him   geometry   concisely,  but  Menaechmus  replied:  "O  king, 
through  the  country  there  are  royal  roads  and  roads  for  common 
citizens,  but  in  geometry  there  is  one  road  for  all." 

Stobceus  (Edition  Wachsmuth,  Berlin,  1884), 
Ed.  2,  p.  80. 

902.  Alexander  the  king  of  the  Macedonians,  began  like  a 
wretch  to  learn  geometry,  that  he  might  know  how  little  the 
earth  was,  whereof  he  had  possessed  very  little.    Thus,  I  say, 
like  a  wretch  for  this,  because  he  was  to  understand  that  he 
did  bear  a  false  surname.    For  who  can  be  great  in  so  small  a 
thing?    Those  things  that  were  delivered  were  subtile,  and  to  be 
learned  by  diligent  attention:  not  which  that  mad  man  could 
perceive,  who  sent  his  thoughts  beyond  the  ocean  sea.    Teach 
me,  saith  he,  easy  things.     To  whom  his  master  said:  These 
things  be  the  same,  and  alike  difficult  unto  all.    Think  thou 
that  the  nature  of  things  saith  this.    These  things  whereof  thou 
complainest,  they  are  the  same  unto  all:  more  easy  things  can 
be  given  unto  none;  but  whosoever  will,  shall  make  those  things 
more  easy  unto  himself.     How?  With  uprightness  of  mind. 

SENECA. 
Epistle  91  [Thomas  Lodge]. 

903.  Archimedes  .  .  .  had  stated  that  given  the  force,  any 
given  weight  might  be  moved,  and  even  boasted,  we  are  told, 
relying  on  the  strength  of  demonstration,  that  if  there  were 
another  earth,  by  going  into  it  he  could  remove  this.    Hiero 
being  struck  with  amazement  at  this,  and  entreating  him  to 
make  good  this  problem  by  actual  experiment,  and  show  some 
great  weight  moved  by  a  small  engine,  he  fixed  accordingly  upon 
a  ship  of  burden  out  of  the  king's  arsenal,  which  could  not  be 
drawn  out  of  the  dock  without  great  labor  and  many  men;  and, 

135 


136  MEMORABILIA   MATHEMATICA 

loading  her  with  many  passengers  and  a  full  freight,  sitting  him- 
self the  while  far  off  with  no  great  endeavor,  but  only  holding 
the  head  of  the  pulley  in  his  hand  and  drawing  the  cords  by 
degrees,  he  drew  the  ship  in  a  straight  line,  as  smoothly  and 
evenly,  as  if  she  had  been  in  the  sea.  The  king,  astonished  at 
this,  and  convinced  of  the  power  of  the  art,  prevailed  upon 
Archimedes  to  make  him  engines  accommodated  to  all  the  pur- 
poses, offensive  and  defensive,  of  a  siege.  .  .  .  the  apparatus 
was,  in  most  opportune  time,  ready  at  hand  for  the  Syracusans, 
and  with  it  also  the  engineer  himself. — PLUTARCH. 

Life  of  Marcellus  [Dry den]. 

904.  These  machines  [used  in  the  defense  of  the  Syracusans 
against  the  Romans  under  Marcellus]  he  [Archimedes]  had 
designed  and  contrived,  not  as  matters  of  any  importance,  but 
as  mere  amusements  in  geometry;  in  compliance  with  king 
Hiero's  desire  and  request,  some  time  before,  that  he  should 
reduce  to  practice  some  part  of  his  admirable  speculation  in 
science,  and  by  accommodating  the  theoretic  truth  to  sensation 
and  ordinary  use,  bring  it  more  within  the  appreciation  of 
people  in  general.  Eudoxus  and  Archytas  had  been  the  first 
originators  of  this  far-famed  and  highly-prized  art  of  mechanics, 
which  they  employed  as  an  elegant  illustration  of  geometrical 
truths,  and  as  means  of  sustaining  experimentally,  to  the 
satisfaction  of  the  senses,  conclusions  too  intricate  for  proof  by 
words  and  diagrams.  As,  for  example,  to  solve  the  problem,  so 
often  required  in  constructing  geometrical  figures,  given  the 
two  extremes,  to  find  the  two  mean  lines  of  a  proportion,  both 
these  mathematicians  had  recourse  to  the  aid  of  instruments, 
adapting  to  their  purpose  certain  curves  and  sections  of  lines. 
But  what  with  Plato's  indignation  at  it,  and  his  invectives 
against  it  as  the  mere  corruption  and  annihilation  of  the  one 
good  of  geometry, — which  was  thus  shamefully  turning  its  back 
upon  the  unembodied  objects  of  pure  intelligence  to  recur  to 
sensation,  and  to  ask  help  (not  to  be  obtained  without  base 
supervisions  and  depravation)  from  matter;  so  it  was  that 
mechanics  came  to  be  separated  from  geometry,  and,  repudiated 
and  neglected  by  philosophers,  took  its  place  as  a  military  art. 

PLUTARCH. 
Life  of  Marcellus  [Dry den]. 


PERSONS  AND  ANECDOTES  137 

905.  Archimedes  was  not  free  from  the  prevailing  notion  that 
geometry  was  degraded  by  being  employed  to  produce  anything 
useful.  It  was  with  difficulty  that  he  was  induced  to  stoop  from 
speculation  to  practice.  He  was  half  ashamed  of  those  inven- 
tions which  were  the  wonder  of  hostile  nations,  and  always 
spoke  of  them  slightingly  as  mere  amusements,  as  trifles  in 
which  a  mathematician  might  be  suffered  to  relax  his  mind  after 
intense  application  to  the  higher  parts  of  his  science. 

MACAULAY. 

Lord  Bacon;  Edinburgh  Review,  July  1837; 
Critical  and  Miscellaneous  Essays  (New  York, 
1879),  Vol.  1,  p.  880. 


906.  Call  Archimedes  from  his  buried  tomb 
Upon  the  plain  of  vanished  Syracuse, 
And  feelingly  the  sage  shall  make  report 
How  insecure,  how  baseless  in  itself, 
Is  the  philosophy,  whose  sway  depends 
On  mere  material  instruments — how  weak 
Those  arts,  and  high  inventions,  if  unpropped 
By  virtue. — WORDSWORTH. 

The  Excursion. 


907.  Zu  Archimedes  kam  einst  ein  wissbegieriger  Jungling. 
"Weihe  mich,"  sprach  er  zu  ihm,  "ein  in  die  gottliche 

Kunst, 

Die  so  herrliche  Frucht  dem  Vaterlande  getragen, 
Und    die    Mauern  der  Stadt  vor   der  Sambuca   bes- 

chiitzt!" 
"  Gottlich  nennst  du  die  Kunst?    Sie  ists,"  versetzte  der 

Weise; 
"Aber  das  war  sie,  mein  Sohn,  eh  sie  dem  Staat  noch 

gedient. 
Willst  du  nur  Friichte  von  ihr,  die  kann  auch  die  Ster- 

bliche  zeugen; 

Wer  urn  die  Gottin  freit,  suche  in  ihr  nicht  das  Weib." 

SCHILLER. 
Archimedes  und  der  Schuler. 


138  MEMORABILIA    MATHEMATICA 

[To  Archimedes  once  came  a  youth  intent  upon  knowl- 
edge. 

Said  he  "Initiate  me  into  the  Science  divine, 

Which  to  our  country  has  borne  glorious  fruits  in  abund- 
ance, 

And  which  the  walls  of  the  town  'gainst  the  Sambuca 
protects." 

"Callst  thou  the  science  divine?  It  is  so,"  the  wise  man 
responded; 

"But  so  it  was,  my  son,  ere  the  state  by  her  service  was 
blest. 

Would'st  thou  have  fruit  of  her  only?  Mortals  with  that 
can  provide  thee, 

He  who  the  goddess  would  woo,  seek  not  the  woman  in 
her."] 

908.  Archimedes  possessed  so  high  a  spirit,  so  profound  a 
soul,  and  such  treasures  of  highly  scientific  knowledge,  that 
though  these  inventions  [used  to  defend  Syracuse  against  the 
Romans]  had  now  obtained  him  the  renown  of  more  than  human 
sagacity,  he  yet  would  not  deign  to  leave  behind  him  any  com- 
mentary or  writing  on  such  subjects;  but,  repudiating  as  sordid 
and  ignoble  the  whole  trade  of  engineering,  and  every  sort  of 
art  that  lends  itself  to  mere  use  and  profit,  he  placed  his  whole 
affection  and  ambition  in  those  purer  speculations  where  there 
can  be  no  reference  to  the  vulgar  needs  of  life;  studies,  the 
superiority  of  which  to  all  others  is  unquestioned,  and  in  which 
the  only  doubt  can  be  whether  the  beauty  and  grandeur  of  the 
subjects  examined,  or  the  precision  and  cogency  of  the  methods 
and  means  of  proof,  most  deserve  our  admiration. — PLUTARCH. 

Life  of  Marcellus  [Dryden]. 

909.  Nothing  afflicted  Marcellus  so  much  as  the  death  of 
Archimedes,  who  was  then,  as  fate  would  have  it,  intent  upon 
working  out  some  problem  by  a  diagram,  and  having  fixed  his 
mind  alike  and  his  eyes  upon  the  subject  of  his  speculation,  he 
never  noticed  the  incursion  of  the  Romans,  nor  that  the  city  was 
taken.    In  this  transport  of  study  and  contemplation,  a  soldier, 
unexpectedly  coming  up  to  him,  commanded  him  to  follow  to 


PERSONS  AND  ANECDOTES  139 

Marcellus,  which  he  declined  to  do  before  he  had  worked  out 
his  problem  to  a  demonstration;  the  soldier,  enraged,  drew  his 
sword  and  ran  him  through.  Others  write,  that  a  Roman 
soldier,  running  upon  him  with  a  drawn  sword,  offered  to  kill 
him;  and  that  Archimedes,  looking  back,  earnestly  besought 
him  to  hold  his  hand  a  little  while,  that  he  might  not  leave  what 
he  was  at  work  upon  inconclusive  and  imperfect;  but  the  soldier, 
nothing  moved  by  his  entreaty,  instantly  killed  him.  Others 
again  relate,  that  as  Archimedes  was  carrying  to  Marcellus 
mathematical  instruments,  dials,  spheres,  and  angles,  by  which 
the  magnitude  of  the  sun  might  be  measured  to  the  sight,  some 
soldiers  seeing  him,  and  thinking  that  he  carried  gold  in  a 
vessel,  slew  him.  Certain  it  is,  that  his  death  was  very  afflicting 
to  Marcellus;  and  that  Marcellus  ever  after  regarded  him  that 
killed  him  as  a  murderer;  and  that  he  sought  for  his  kindred  and 
honoured  them  with  signal  favours. — PLUTARCH. 

Life  of  Marcellus  [Dryden]. 

910.  [Archimedes]  is  said  to  have  requested  his  friends  and 
relations  that  when  he  was  dead,  they  would  place  over  his 
tomb  a  sphere  containing  a  cylinder,  inscribing  it  with  the  ratio 
which  the  containing  solid  bears  to  the  contained. — PLUTARCH. 

Life  of  Marcellus  [Dryden]. 

911.  Archimedes,  who  combined  a  genius  for  mathematics 
with  a  physical  insight,  must  rank  with  Newton,  who  lived 
nearly  two  thousand  years  later,  as  one  of  the  founders  of  math- 
ematical physics.  .  .  .     The  day  (when  having  discovered  his 
famous  principle  of  hydrostatics  he  ran  through  the  streets 
shouting  Eureka!  Eureka!)  ought  to  be  celebrated  as  the  birth- 
day of  mathematical  physics;  the  science  came  of  age  when 
Newton  sat  in  his  orchard. — WHITEHEAD,  A.  N. 

An  Introduction  to  Mathematics  (New  York, 
1911),  p.  38. 

912.  It  is  not  possible  to  find  in  all  geometry  more  difficult 
and  more  intricate  questions  or  more  simple  and  lucid  explana- 
tions [than  those  given  by  Archimedes].    Some  ascribe  this  to 
his  natural  genius;  while  others  think  that  incredible  effort  and 
toil  produced  these,  to  all  appearance,  easy  and  unlaboured 


140  MEMORABILIA   MATHEMATICA 

results.  No  amount  of  investigation  of  yours  would  succeed  in 
attaining  the  proof,  and  yet,  once  seen,  you  immediately  believe 
you  would  have  discovered  it;  by  so  smooth  and  so  rapid  a  path 
he  leads  you  to  the  conclusion  required. — PLUTARCH. 

Life  of  Marcellus  [Dryden]. 

913.  One  feature  which  will  probably  most  impress  the 
mathematician  accustomed  to  the  rapidity  and  directness 
secured  by  the  generality  of  modern  methods  is  the  deliberation 
with  which  Archimedes  approaches  the  solution  of  any  one  of 
his  main  problems.  Yet  this  very  characteristic,  with  its 
incidental  effects,  is  calculated  to  excite  the  more  admiration 
because  the  method  suggests  the  tactics  of  some  great  strategist 
who  foresees  everything,  eliminates  everything  not  immediately 
conducive  to  the  execution  of  his  plan,  masters  every  position 
in  its  order,  and  then  suddenly  (when  the  very  elaboration  of  the 
scheme  has  almost  obscured,  in  the  mind  of  the  spectator,  its 
ultimate  object)  strikes  the  final  blow.  Thus  we  read  in  Arch- 
imedes proposition  after  proposition  the  bearing  of  which  is  not 
immediately  obvious  but  which  we  find  infallibly  used  later 
on;  and  we  are  led  by  such  easy  stages  that  the  difficulties  of  the 
original  problem,  as  presented  at  the  outset,  are  scarcely  appre- 
ciated. As  Plutarch  says :  "  It  is  not  possible  to  find  in  geometry 
more  difficult  and  troublesome  questions,  or  more  simple  and 
lucid  explanations."  But  it  is  decidedly  a  rhetorical  exaggera- 
tion when  Plutarch  goes  on  to  say  that  we  are  deceived  by  the 
easiness  of  the  successive  steps  into  the  belief  that  anyone  could 
have  discovered  them  for  himself.  On  the  contrary,  the  studied 
simplicity  and  the  perfect  finish  of  the  treatises  involve  at  the 
same  time  an  element  of  mystery.  Though  each  step  depends 
on  the  preceding  ones,  we  are  left  in  the  dark  as  to  how  they 
were  suggested  to  Archimedes.  There  is,  in  fact,  much  truth  in 
a  remark  by  Wallis  to  the  effect  that  he  seems  "as  it  were  of  set 
purpose  to  have  covered  up  the  traces  of  his  investigation  as  if 
he  had  grudged  posterity  the  secret  of  his  method  of  inquiry 
while  he  wished  to  extort  from  them  assent  to  his  results." 
Wallis  adds  with  equal  reason  that  not  only  Archimedes  but 
nearly  all  the  ancients  so  hid  away  from  posterity  their  method 
of  Analysis  (though  it  is  certain  that  they  had  one)  that  more 


PERSONS  AND  ANECDOTES  141 

modern  mathematicians  found  it  easier  to  invent  a  new  Analysis 
than  to  seek  out  the  old. — HEATH,  T.  L. 

The  Works  of  Archimedes  (Cambridge,  1897), 
Preface. 

914.  It  is  a  great  pity  Aristotle  had  not  understood  mathe- 
matics as  well  as  Mr.  Newton,  and  made  use  of  it  in  his  natural 
philosophy  with  good  success :  his  example  had  then  authorized 
the  accommodating  of  it  to  material  things. — LOCKE,  JOHN. 

Second  Reply  to  the  Bishop  of  Worcester. 

915.  The  opinion  of  Bacon  on  this  subject  [geometry]  was 
diametrically  opposed  to  that  of  the  ancient  philosophers.    He 
valued  geometry  chiefly,  if  not  solely,  on  account  of  those  uses, 
which  to  Plato  appeared  so  base.    And  it  is  remarkable  that  the 
longer  Bacon  lived  the  stronger  this  feeling  became.    When  in 
1605  he  wrote  the  two  books  on  the  Advancement  of  Learning, 
he  dwelt  on  the  advantages  which  mankind  derived  from  mixed 
mathematics;  but  he  at  the  same  time  admitted  that  the  benefi- 
cial effect  produced  by  mathematical  study  on  the  intellect, 
though  a  collateral  advantage,  was  "no  less  worthy  than  that 
which  was  principal  and  intended."    But  it  is  evident  that  his 
views  underwent  a  change.    When  near  twenty  years  later,  he 
published  the  De  Augmentis,  which  is  the  Treatise  on  the  Ad- 
vancement of  Learning,  greatly  expanded  and  carefully  cor- 
rected, he  made  important  alterations  in  the  part  which  related 
to  mathematics.    He  condemned  with  severity  the  pretensions 
of  the  mathematicians,  "delidas  et  fastum  mathematicorum." 
Assuming  the  well-being  of  the  human  race  to  be  the  end  of 
knowledge,  he  pronounced  that  mathematical  science  could 
claim  no  higher  rank  than  that  of  an  appendage  or  an  auxiliary 
to  other  sciences.    Mathematical  science,  he  says,  is  the  hand- 
maid of  natural  philosophy;  she  ought  to  demean  herself  as  such; 
and  he  declares  that  he  cannot  conceive  by  what  ill  chance  it  has 
happened  that  she  presumes  to  claim  precedence  over  her 
mistress. — MACAULAY. 

Lord  Bacon:  Edinburgh  Review,  July,  1837; 
Critical  and  Miscellaneous  Essays  (New  York, 
1879),  Vol.  1,  p.  380. 


142  MEMORABILIA   MATHEMATICA 

916.  If  Bacon  erred  here  [in  valuing  mathematics  only  for  its 
uses],  we  must  acknowledge  that  we  greatly  prefer  his  error  to 
the  opposite  error  of  Plato.    We  have  no  patience  with  a  philoso- 
phy which,  like  those  Roman  matrons  who  swallowed  abortives 
in  order  to  preserve  their  shapes,  takes  pains  to  be  barren  for 
fear  of  being  homely. — MACAULAY. 

Lord  Bacon,  Edinburgh  Review,  July,  1887; 
Critical  and  Miscellaneous  Essays  (New  York, 
1879),  Vol.  2,  p.  381. 

917.  He  [Lord  Bacon]  appears  to  have  been  utterly  ignorant 
of  the  discoveries  which  had  just  been  made  by  Kepler's  calcula- 
tions ...  he  does  not  say  a  word  about  Napier's  Logarithms, 
which  had  been  published  only  nine  years  before  and  reprinted 
more  than  once  in  the  interval.    He  complained  that  no  con- 
siderable advance  had  been  made  in  Geometry  beyond  Euclid, 
without  taking  any  notice  of  what  had  been  done  by  Archimedes 
and  Apollonius.     He  saw  the  importance  of  determining  ac- 
curately the  specific  gravities  of  different  substances,  and  him- 
self attempted  to  form  a  table  of  them  by  a  rude  process  of  his 
own,  without  knowing  of  the  more  scientific  though  still  imper- 
fect methods  previously  employed  by  Archimedes,  Ghetaldus 
and   Porta.      He  speaks  of  the  evpijica  of  Archimedes   in   a 
manner  which  implies  that  he  did  not  clearly  appreciate  either 
the  problem  to  be  solved  or  the  principles  upon  which  the 
solution  depended.     In  reviewing  the  progress  of  Mechanics, 
he  makes  no  mention  either  of  Archimedes,  or  Stevinus,  Galileo, 
Guldinus,  or  Ghetaldus.    He  makes  no  allusion  to  the  theory  of 
Equilibrium.    He  observes  that  a  ball  of  one  pound  weight  will 
fall  nearly  as  fast  through  the  air  as  a  ball  of  two,  without  allud- 
ing to  the  theory  of  acceleration  of  falling  bodies,  which  had 
been  made  known  by  Galileo  more  than  thirty  years  before. 
He  proposed  an  inquiry  with  regard  to  the  lever, — namely, 
whether  in  a  balance  with  arms  of  different  length  but  equal 
weight  the  distance  from  the  fulcrum  has  any  effect  upon  the 
inclination — though  the  theory  of  the  lever  was  as  well  under- 
stood in  his  own  time  as  it  is  now.  .  .  .  He  speaks  of  the  poles 
of  the  earth  as  fixed,  in  a  manner  which  seems  to  imply  that 
he  was  not  acquainted  with  the  precession  of  the  equinoxes; 
and  in  another  place,  of  the  north  pole  being  above  and  the 


PERSONS   AND   ANECDOTES  143 

south  pole  below,  as  a  reason  why  in  our  hemisphere  the  north 
winds  predominate  over  the  south. — SPEDDING,  J. 

Works  of  Francis  Bacon  (Boston),  Preface  to 
De  Interpretation  Naturae  Prooemium. 

918.  Bacon  himself  was  very  ignorant  of  all  that  had  been 
done  by  mathematics;  and,  strange  to  say,  he  especially  ob- 
jected to  astronomy  being  handed  over  to  the  mathematicians. 
Leverrier  and  Adams,  calculating  an  unknown  planet  into  a 
visible  existence  by  enormous  heaps  of  algebra,  furnish  the  last 
comment  of  note  on  this  specimen  of  the  goodness  of  Bacon's 
view.  .  .  .  Mathematics  was  beginning  to  be  the  great  instru- 
ment of  exact  inquiry :  Bacon  threw  the  science  aside,  from  igno- 
rance, just  at  the  time  when  his  enormous  sagacity,  applied  to 
knowledge,  would  have  made  him  see  the  part  it  was  to  play. 
If  Newton  had  taken  Bacon  for  his  master,  not  he,  but  some- 
body else,  would  have  been  Newton. — DE  MORGAN,  A. 

Budget  of  Paradoxes  (London,  1872},  pp.  53-54- 

919.  Daniel  Bernoulli  used  to  tell  two  little  adventures,  which 
he  said  had  given  him  more  pleasure  than  all  the  other  honours 
he  had  received.    Travelling  with  a  learned  stranger,  who,  being 
pleased  with  his  conversation,  asked  his  name;  "I  am  Daniel 
Bernoulli,"  answered  he  with  great  modesty;  "and  I,"  said  the 
stranger  (who  thought  he  meant  to  laugh  at  him)  "am  Isaac 
Newton."    Another  time,  having  to  dine  with  the  celebrated 
Koenig,  the  mathematician,  who  boasted,  with  some  degree  of 
self-complacency,  of  a  difficult  problem  he  had  solved  with 
much  trouble,  Bernoulli  went  on  doing  the  honours  of  his  table, 
and  when  they  went  to  drink  coffee  he  presented  Koenig  with  a 
solution  of  the  problem  more  elegant  than  his  own. 

HUTTON,  CHARLES. 

A  Philosophical  and  Mathematical  Dictionary 
(London,  1815),  Vol.  1,  p.  226. 

920.  Following  the  example  of  Archimedes  who  wished  his 
tomb  decorated  with  his  most  beautiful  discovery  in  geometry 
and  ordered  it  inscribed  with  a  cylinder  circumscribed  by  a 
sphere,  James  Bernoulli  requested  that  his  tomb  be  inscribed 
with  his  logarithmic  spiral  together  with  the  words,  "Eadem 


144  MEMORABILIA   MATHEMATICA 

mutata  resurgo,"  a  happy  allusion  to  the  hope  of  the  Christians, 
which  is  in  a  way  symbolized  by  the  properties  of  that  curve. 

FONTENELLE. 

Eloge  de  M.  Bernoulli;  Oeuvres  de  Fontenelle, 
t.  6  (1768),  p.  112. 

921.  This  formula  [for  computing  Bernoulli's  numbers]  was 
first  given  by  James  Bernoulli.     He  gave  no  general  demon- 
stration; but  was  quite  aware  of  the  importance  of  his  theorem, 
for  he  boasts  that  by  means  of  it  he  calculated  intra  semi- 
quadrantem  horae!  the  sum  of  the   10th  powers  of   the   first 
thousand  integers,  and  found  it  to  be 

91,409,924,241,424,243,424,241,924,242,500. 

CHRYSTAL,  G. 
Algebra,  Part  2  (Edinburgh,  1879) ,  p.  209. 

922.  In  the  year  1692,  James  Bernoulli,  discussing  the  log- 
arithmic spiral  [or  equiangular  spiral,  p  =  a  0  ]  .  .  .  shows  that 
it  reproduces  itself  in  its  evolute,  its  involute,  and  its  caustics  of 
both  reflection  and  refraction,  and  then  adds:  "But  since  this 
marvellous  spiral,  by  such  a  singular  and  wonderful  peculiarity, 
pleases  me  so  much  that  I  can  scarce  be  satisfied  with  thinking 
about  it,  I  have  thought  that  it  might  not  be  inelegantly  used 
for  a  symbolic  representation  of  various  matters.    For  since  it 
always  produces  a  spiral  similar  to  itself,  indeed  precisely  the 
same  spiral,  however  it  may  be  involved  or  evolved,  or  reflected 
or  refracted,  it  may  be  taken  as  an  emblem  of  a  progeny  always 
in  all  things  like  the  parent,  simillima  filia  matri.    Or,  if  it  is  not 
forbidden  to  compare  a  theorem  of  eternal  truth  to  the  myste- 
ries of  our  faith,  it  may  be  taken  as  an  emblem  of  the  eternal 
generation  of  the  Son,  who  as  an  image  of  the  Father,  emanating 
from  him,  as  light  from  light,  remains  o/ioou<rto?  with  him, 
howsoever  overshadowed.     Or,  if  you  prefer,  since  our  spira 
mirabilis  remains,  amid  all  changes,  most  persistently  itself, 
and  exactly  the  same  as  ever,  it  may  be  used  as  a  symbol,  either 
of  fortitude  and  constancy  in  adversity,  or,  of  the  human  body, 
which  after  all  its  changes,  even  after  death,  will  be  restored  to 
its  exact  and  perfect  self,  so  that,  indeed,  if  the  fashion  of 
Archimedes  were  allowed  in  these  days,  I  should  gladly  have  my 


PERSONS   AND   ANECDOTES  145 

tombstone  bear  this  spiral,  with  the  motto,  "  Though  changed, 
I  arise  again  exactly  the  same,  Eadem  numero  mutata  resurgo." 

HILL,  THOMAS. 

The  Uses  of  Mathesis;  Bibliotheca  Sacra,  Vol. 

82,  pp.  515-516. 

923.  Babbage  was  one  of  the  founders  of  the  Cambridge 
Analytical  Society  whose  purpose  he  stated  was  to  advocate 
"the  principles  of  pure  d-ism  as  opposed  to  the  dof-age  of  the 
university."— BALL,  W.  W.  R. 

History  of  Mathematics  (London,  1901),  p.  451. 

924.  Bolyai  [Janos]  when  in  garrison  with  cavalry  officers, 
was  provoked  by  thirteen  of  them  and  accepted  all  their  chal- 
lenges on  condition  that  he  be  permitted  after  each  duel  to  play 
a  bit  on  his  violin.    He  came  out  victor  from  his  thirteen  duels, 
leaving  his  thirteen  adversaries  on  the  square. — HALSTED.  G.  B. 

Bolyai' s  Science  Absolute  of  Space  (Austin, 
1896),  Introduction,  p.  29. 

925.  Bolyai  [Janos]  projected  a  universal  language  for  speech 
as  we  have  it  for  music  and  mathematics. — HALSTED,  G.  B. 

Bolyai's  Science  Absolute  of  Space  (Austin, 
1896),  Introduction,  p.  29. 

926.  [Bolyai's   Science  Absolute  of   Space] — the  most   ex- 
traordinary two  dozen  pages  in  the  history  of  thought! 

HALSTED,  G.  B. 

Bolyai's  Science  Absolute  of  Space  (Austin, 
1896),  Introduction,  p.  18. 

927.  [Wolfgang  Bolyai]  was  extremely  modest.     No  monu- 
ment, said  he,  should  stand  over  his  grave,  only  an  apple-tree, 
in  memory  of  the  three  apples :  the  two  of  Eve  and  Paris,  which 
made  hell  out  of  earth,  and  that  of  Newton,  which  elevated  the 
earth  again  into  the  circle  of  the  heavenly  bodies. — CAJOKI,  F. 

History  of  Elementary  Mathematics  (New  York, 
1910),  p.  273. 

928.  Bernard  Bolzano  dispelled  the  clouds  that  throughout 
all  the  foregone  centuries  had  enveloped  the  notion  of  Infinitude 


146  MEMORABILIA   MATHEMATICA 

in  darkness,  completely  sheared  the  great  term  of  its  vagueness 
without  shearing  it  of  its  strength,  and  thus  rendered  it  forever 
available  for  the  purposes  of  logical  discourse. — KEYSER,  C.  J. 

Lectures  on  Science,  Philosophy  and  Art  (New 

York,  1908),  p.  42. 

929.  Let  me  tell  you  how  at  one  time  the  famous  mathemati- 
cian Euclid  became  a  physician.     It  was  during  a  vacation, 
which  I  spent  in  Prague  as  I  most  always  did,  when  I  was  at- 
tacked by  an  illness  never  before  experienced,  which  mani- 
fested itself  hi  chilliness  and  painful  weariness  of  the  whole 
body.    In  order  to  ease  my  condition  I  took  up  Euclid's  Ele- 
ments and  read  for  the  first  tune  his  doctrine  of  ratio,  which  I 
found  treated  there  in  a  manner  entirely  new  to  me.     The 
ingenuity  displayed  in  Euclid's  presentation  filled  me  with  such 
vivid  pleasure,  that  forthwith  I  felt  as  well  as  ever. 

BOLZANO,  BERNARD. 

Selbstbiographie  (Wien,  1875),  p.  20. 

930.  Mr.  Cayley,  of  whom  it  may  be  so  truly  said,  whether 
the  matter  he  takes  in  hand  be  great  or  small,  "nihil  tetigit  quod 
non  ornavit"  .  .  .  — SYLVESTER,  J.  J. 

Philosophic  Transactions  of  the  Royal  Society, 
Vol.  17  (1864),  P-  605. 

931.  It  is  not  Cayley's  way  to  analyze  concepts  into  their 
ultimate  elements.  .  .  .     But  he  is  master  of  the  empirical 
utilization  of  the  material :  in  the  way  he  combines  it  to  form  a 
single  abstract  concept  which  he  generalizes  and  then  subjects 
to  computative  tests,  in  the  way  the  newly  acquired  data  are 
made  to  yield  at  a  single  stroke  the  general  comprehensive  idea 
to  the  subsequent  numerical  verification  of  which  years  of 
labor  are  devoted.    Cayley  is  thus  the  natural  philosopher  among 
mathematicians. — NOETHER,  M. 

Mathematische  Annakn,  Bd.  46  (1895),  p.  479. 

932.  When  Cayley  had  reached  his  most  advanced  generaliza- 
tions he  proceeded  to  establish  them  directly  by  some  method  or 
other,  though  he  seldom  gave  the  clue  by  which  they  had  first 
been  obtained:  a  proceeding  which  does  not  tend  to  make  his 
papers  easy  reading.  .  .  . 


PERSONS   AND   ANECDOTES  147 

His  literary  style  is  direct,  simple  and  clear.  His  legal  train- 
ing had  an  influence,  not  merely  upon  his  mode  of  arrangement 
but  also  upon  his  expression;  the  result  is  that  his  papers  are 
severe  and  present  a  curious  contrast  to  the  luxuriant  enthusiasm 
which  pervades  so  many  of  Sylvester's  papers.  He  used  to 
prepare  his  work  for  publication  as  soon  as  he  carried  his  inves- 
tigations in  any  subject  far  enough  for  his  immediate  pur- 
pose. ...  A  paper  once  written  out  was  promptly  sent  for 
publication;  this  practice  he  maintained  throughout  life.  .  .  . 
The  consequence  is  that  he  has  left  few  arrears  of  unfinished  or 
unpublished  papers;  his  work  has  been  given  by  himself  to  the 
world. — FORSYTH,  A.  R. 

Proceedings  of  London  Royal  Society,  Vol.  58 

(1895),  pp.  23-24. 

933.  Cayley  was  singularly  learned  in  the  work  of  other  men, 
and  catholic  in  his  range  of  knowledge.    Yet  he  did  not  read  a 
memoir  completely  through:  his  custom  was  to  read  only  so 
much  as  would  enable  him  to  grasp  the  meaning  of  the  symbols 
and  understand  its  scope.    The  main  result  would  then  become 
to  him  a  subject  of  investigation:  he  would  establish  it  (or  test 
it)  by  algebraic  analysis  and,  not  infrequently,  develop  it  so  to 
obtain  other  results.     This  faculty  of  grasping  and  testing 
rapidly  the  work  of  others,  together  with  his  great  knowledge, 
made  him  an  invaluable  referee;  his  services  in  this  capacity 
were  used  through  a  long  series  of  years  by  a  number  of  societies 
to  which  he  was  almost  in  the  position  of  standing  mathematical 
advisor. — FORSYTH,  A.  R. 

Proceedings  London  Royal  Society,   Vol.  68 
(1895),  pp.  11-12. 

934.  Bertrand,    Darboux,    and    Glaisher    have    compared 
Cayley  to  Euler,  alike  for  his  range,  his  analytical  power,  and, 
not  least,  for  his  prolific  production  of  new  views  and  fertile 
theories.    There  is  hardly  a  subject  in  the  whole  of  pure  mathe- 
matics at  which  he  has  not  worked. — FORSYTH,  A.  R. 

Proceedings  London  Royal  Society,   Vol.  58 
(1895),  p.  21. 

935.  The  mathematical  talent  of  Cayley  was  characterized  by 
clearness  and  extreme  elegance  of  analytical  form;  it  was  re- 


148  MEMORABILIA   MATHEMATICA 

enforced  by  an  incomparable  capacity  for  work  which  has  caused 
the  distinguished  scholar  to  be  compared  with  Cauchy. 

HERMITE,  C. 

Comptes  Rendus,  1. 120  (1895),  p.  284. 

936.  J.  J.  Sylvester  was  an  enthusiastic  supporter  of  reform 
[in  the  teaching  of  geometry].    The  difference  in  attitude  on  this 
question  between  the  two  foremost  British  mathematicians, 
J.  J.  Sylvester,  the  algebraist,  and  Arthur  Cayley,  the  alge- 
braist and  geometer,  was  grotesque.    Sylvester  wished  to  bury 
Euclid  "deeper  than  e'er  plummet  sounded"  out  of  the  school- 
boy's reach;  Cayley,  an  ardent  admirer  of  Euclid,  desired  the 
retention  of  Simson's  Euclid.    When  reminded  that  this  treatise 
was  a  mixture  of  Euclid  and  Simson,  Cayley  suggested  striking 
out  Simson's  additions  and  keeping  strictly  to  the  original 
treatise. — CAJORI,  F. 

History    of   Elementary    Mathematics    (New 
York,  1910),  p.  285. 

937.  Tait  once  urged  the  advantage  of  Quaternions  on  Cayley 
(who  never  used  them),  saying:  "You  know  Quaternions  are 
just  like  a  pocket-map."     "That  may  be,"  replied  Cayley, 
"but  you've  got  to  take  it  out  of  your  pocket,  and  unfold  it, 
before  it's  of  any  use."    And  he  dismissed  the  subject  with  a 
smile. — THOMPSON,  S.  P. 

Life  of  Lord  Kelvin  (London,  1910),  p.  1137. 

938.  As  he  [Clifford]  spoke  he  appeared  not  to  be  working  out 
a  question,  but  simply  telling  what  he  saw.     Without  any 
diagram  or  symbolic  aid  he  described  the  geometrical  conditions 
on  which  the  solution  depended,  and  they  seemed  to  stand  out 
visibly  in  space.    There  were  no  longer  consequences  to  be  de- 
duced, but  real  and  evident  facts  which  only  required  to  be 
seen.  ...    So  whole  and  complete  was  his  vision  that  for  the 
time  the  only  strange  thing  was  that  anybody  should  fail  to  see 
it  in  the  same  way.    When  one  endeavored  to  call  it  up  again, 
and  not  till  then,  it  became  clear  that  the  magic  of  genius  had 
been  at  work,  and  that  the  common  sight  had  been  raised  to  that 
higher  perception  by  the  power  that  makes  and  transforms 


PERSONS  AND  ANECDOTES  149 

ideas,  the  conquering  and  masterful  quality  of  the  human  mind 
which  Goethe  called  in  one  word  das  Ddmonische. — POLLOCK,  F. 

Clifford's  Lectures  and  Essays   (New   York, 
1901),  Vol.  1,  Introduction,  pp.  5-6. 

939.  Much  of  his  [Clifford's]  best  work  was  actually  spoken 
before  it  was  written.    He  gave  most  of  his  public  lectures  with 
no  visible  preparation  beyond  very  short  notes,  and  the  outline 
seemed  to  be  filled  in  without  effort  or  hesitation.    Afterwards 
he  would  revise  the  lecture  from  a  shorthand  writer's  report,  or 
sometimes  write  down  from  memory  almost  exactly  what  he  had 
said.    It  fell  out  now  and  then,  however,  that  neither  of  these 
things  was  done;  in  such  cases  there  is  now  no  record  of  the 
lecture  at  all. — POLLOCK,  F. 

Clifford's  Lectures  and  Essays  (New   York, 
1901),  Vol.  1,  Introduction,  p.  10. 

940.  I  cannot  find  anything  showing  early  aptitude  for  ac- 
quiring languages;  but  that  he  [Clifford]  had  it  and  was  fond  of 
exercising  it  in  later  life  is  certain.    One  practical  reason  for  it 
was  the  desire  of  being  able  to  read  mathematical  papers  in 
foreign  journals;  but  this  would  not  account  for  his  taking  up 
Spanish,  of  which  he  acquired  a  competent  knowledge  in  the 
course  of  a  tour  to  the  Pyrenees.    When  he  was  at  Algiers  in 
1876  he  began  Arabic,  and  made  progress  enough  to  follow  in  a 
general  way  a  course  of  lessons  given  in  that  language.    He  read 
modern  Greek  fluently,  and  at  one  time  he  was  furious  about 
Sanskrit.    He  even  spent  some  time  on  hieroglyphics.    A  new 
language  is  a  riddle  before  it  is  conquered,  a  power  in  the  hand 
afterwards:  to  Clifford  every  riddle  was  a  challenge,  and  every 
chance  of  new  power  a  divine  opportunity  to  be  seized.    Hence 
he  was  likewise  interested  in  the  various  modes  of  conveying 
and  expressing  language  invented  for  special  purposes,  such  as 
the  Morse  alphabet  and  shorthand.  ...  I  have  forgotten  to 
mention  his  command  of  French  and  German,  the  former  of 
which  he  knew  very  well,  and  the  latter  quite  sufficiently;  .  .  . 

POLLOCK,  F. 

Clifford's  Lectures  and  Essays   (New   York, 
1901),  Vol.  1,  Introduction,  pp.  11-12. 


150  MEMORABILIA   MATHEMATICA 

941.  The  most  remarkable  thing  was  his  [Clifford's]  great 
strength  as  compared  with  his  weight,  as  shown  in  some  exer- 
cises.   At  one  time  he  could  pull  up  on  the  bar  with  either  hand, 
which  is  well  known  to  be  one  of  the  greatest  feats  of  strength. 
His  nerve  at  dangerous  heights  was  extraordinary.    I  am  ap- 
palled now  to  think  that  he  climbed  up  and  sat  on  the  cross 
bars  of  the  weathercock  on  a  church  tower,  and  when  by  way  of 
doing  something  worse  I  went  up  and  hung  by  my  toes  to  the 

bars  he  did  the  same. 

Quoted  from  a  letter  by  one  of  Clifford's  friends 
to  Pollock,  F.:  Clifford's  Lectures  and  Essays 
(New  York,  1901),  Vol.  1,  Introduction,  p.  8. 

942.  [Comte]  may  truly  be  said  to  have  created  the  philoso- 
phy of  higher  mathematics. — MILL,  J.  S. 

System  of  Logic  (New  York,  1846),  p.  369. 

943.  These  specimens,  which  I  could  easily  multiply,  may 
suffice  to  justify  a  profound  distrust  of  Auguste  Comte,  wherever 
he  may  venture  to  speak  as  a  mathematician.    But  his  vast 
general  ability,   and  that  personal  intimacy  with  the  great 
Fourier,  which  I  most  willingly  take  his  own  word  for  having 
enjoyed,  must  always  give  an  interest  to  his  views  on  any  sub- 
ject of  pure  or  applied  mathematics. — HAMILTON,  W.  R. 

Graves'  Life  of  W.  R.  Hamilton  (New  York, 
1882-1889),  Vol.  3,  p.  475. 

944.  The  manner  of  Demoivre's  death  has  a  certain  interest 
for  psychologists.    Shortly  before  it,  he  declared  that  it  was 
necessary  for  him  to  sleep  some  ten  minutes  or  a  quarter  of  an 
hour  longer  each  day  than  the  preceding  one:  the  day  after  he 
had  thus  reached  a  total  of  something  over  twenty-three  hours 
he  slept  up  to  the  limit  of  twenty-four  hours,  and  then  died  in 
his  sleep. — BALL,  W.  W.  R. 

History  of  Mathematics  (London,  1911),  p.  894- 

945.  De  Morgan  was  explaining  to  an  actuary  what  was  the 
chance  that  a  certain  proportion  of  some  group  of  people  would 
at  the  end  of  a  given  time  be  alive;  and  quoted  the  actuarial 
formula,  involving  TT,  which,  in  answer  to  a  question,  he  ex- 
plained stood  for  the  ratio  of  the  circumference  of  a  circle  to  its 


PERSONS  AND  ANECDOTES  151 

diameter.  His  acquaintance,  who  had  BO  far  listened  to  the 
explanation  with  interest,  interrupted  him  and  exclaimed, 
"My  dear  friend,  that  must  be  a  delusion,  what  can  a  circle  have 
to  do  with  the  number  of  people  alive  at  a  given  time?  " 

BALL,  W.  W.  R. 

Mathematical  Recreations  and  Problems  (Lon- 
don, 1896),  p.  180;  See  also  De  Morgan's  Budget 
of  Paradoxes  (London,  1872),  p.  172. 

946.  A  few  days  afterwards,  I  went  to  him  [the  same  actuary 
referred  to  in  945]  and  very  gravely  told  him  that  I  had  dis- 
covered the  law  of  human  mortality  in  the  Carlisle  Table,  of 
which  he  thought  very  highly.    I  told  him  that  the  law  was 
involved  in  this  circumstance.    Take  the  table  of  the  expecta- 
tion of  life,  choose  any  age,  take  its  expectation  and  make  the 
nearest  integer  a  new  age,  do  the  same  with  that,  and  so  on; 
begin  at  what  age  you  like,  you  are  sure  to  end  at  the  place 
where  the  age  past  is  equal,  or  most  nearly  equal,  to  the  ex- 
pectation to  come.     "  You  don't  mean  that  this  always  hap- 
pens?"— "Try  it."    He  did  try,  again  and  again;  and  found  it  as 
I  said.    "  This  is,  indeed,  a  curious  thing;  this  is  a  discovery!" 
I  might  have  sent  him  about  trumpeting  the  law  of  life:  but  I 
contented  myself  with  informing  him  that  the  same  thing  would 
happen  with  any  table  whatsoever  in  which  the  first  column 
goes  up  and  the  second  goes  down;  .  .  . — DE  MORGAN,  A. 

Budget  of  Paradoxes  (London,  1872),  p.  172. 

947.  [De  Morgan  relates  that  some  person  had  made  up  800 
anagrams  on  his  name,  of  which  he  had  seen  about  650.    Com- 
menting on  these  he  says:] 

Two  of  these  I  have  joined  in  the  title-page: 

[Ut  agendo  surgamus  arguendo  gustamus.] 
A  few  of  the  others  are  personal  remarks. 
Great  gun!  do  us  a  sum! 
is  a  sneer  at  my  pursuit;  but,  *,    n 

Go!  great  sum!    /a*  du 
is  more  dignified.  .  .  . 

Adsum,  nugator,  suge! 

is  addressed  to  a  student  who  continues  talking  after  the  lecture 
has  commenced:  . 


152  MEMORABILIA   MATHEMATICA 

Graduatus  sum!  nego 

applies  to  one  who  declined  to  subscribe  for  an  M.  A.  degree. 

DE  MORGAN,  AUGUSTUS. 
Budget  of  Paradoxes  (London,  1872),  p.  82. 

948.  Descartes  is  the  completest  type  which  history  presents 
of  the  purely  mathematical  type  of  mind  —  that  in  which  the 
tendencies  produced  by  mathematical  cultivation  reign  un- 

balanced and  supreme.  —  MILL,  J.  S. 

An  Examination  of  Sir  W.  Hamilton's  Philos- 
ophy (London,  1878),  p.  626. 

949.  To  Descartes,  the  great  philosopher  of  the  17th  century,  is 
due  the  undying  credit  of  having  removed  the  bann  which  until 
then  rested  upon  geometry.    The  analytical  geometry,  as  Des- 
cartes' method  was  called,  soon  led  to  an  abundance  of  new 
theorems  and  principles,  which  far  transcended  everything  that 
ever  could  have  been  reached  upon  the  path  pursued  by  the 

ancients.  —  HANKEL,  H. 

Die   Entwickelung   der   Mathematik   in   den 
letzten  Jahrhunderten  (Tubingen,  1884),  p.  10. 

950.  [The  application  of  algebra  has]  far  more  than  any  of  his 
metaphysical  speculations,  immortalized  the  name  of  Descartes, 
and  constitutes  the  greatest  single  step  ever  made  in  the  progress 

of  the  exact  sciences.  —  MILL,  J.  S. 

An  Examination  of  Sir  W.  Hamilton's  Philos- 
ophy (London,  1878),  p.  617. 

951.  .   .  .  Kai  <j>a<riv  8n  IlToXe/tato?  TJpero  TTOTC  airrov  [Rv/c- 
XeiSrjv],   ei   Ti5    I<TTIV  irepl   'yeajperpiav   68o<j   (rvvTopwrepa   rr)<f 

68%    cnrercplvaTO  f    /LIT;   elvai   j3a<rt\iKr)v   arpcnrbv 


[.  .  .  they  say  that  Ptolemy  once  asked  him  (Euclid)  whether 
there  was  in  geometry  no  shorter  way  than  that  of  the  elements. 
and  he  replied,  "There  is  no  royal  road  to  geometry."] 

PROCLUS. 
(Edition  Friedlein,  1873),  Prol.  II,  39. 

952.  Someone  who  had  begun  to  read  geometry  with  Euclid, 
when  he  had  learned  the  first  proposition,  asked  Euclid,  "But 


PERSONS  AND  ANECDOTES  153 

what  shall  I  get  by  learning  these  things?"  whereupon  Euclid 
called  his  slave  and  said,  "  Give  him  three-pence,  since  he  must 
make  gain  out  of  what  he  learns." — STOB^US. 

(Edition  Wachsmuth,  1884),  Ed.  II 

953.  The  sacred  writings  excepted,  no  Greek  has  been  so 
much  read  and  so  variously  translated  as  Euclid.* 

DE  MORGAN,  A. 

Smith's  Dictionary  of  Greek  and  Roman  Biology 
and  Mythology  (London,  1902),  Article,  "Eu- 
cleides." 

954.  The  thirteen  books  of  Euclid  must  have  been  a  tremen- 
dous advance,  probably  even  greater  than  that  contained  in  the 
"  Principia  "  of  Newton. — DE  MORGAN,  A. 

Smith's  Dictionary  of  Greek  and  Roman  Biog- 
raphy and  Mythology  (London,  1902),  Article, 
"Eucleides." 

955.  To  suppose  that  so  perfect  a  system  as  that  of  Euclid's 
Elements  was  produced  by  one  man,  without  any  preceding 
model  or  materials,  would  be  to  suppose  that  Euclid  was  more 
than  man.   We  ascribe  to  him  as  much  as  the  weakness  of  human 
understanding  will  permit,  if  we  suppose  that  the  inventions  in 
geometry,  which  had  been  made  in  a  tract  of  preceding  ages, 
were  by  him  not  only  carried  much  further,  but  digested  into 
so  admirable  a  system,  that  his  work  obscured  all  that  went 
before  it,  and  made  them  be  forgot  and  lost. — REID,  THOMAS. 

Essay  on  the  Powers  of  the  Human  Mind 
(Edinburgh,  1812),  Vol.  2,  p.  368. 

956.  It  is  the  invaluable  merit  of  the  great  Basle  mathemati- 
cian Leonhard  Euler,  to  have  freed  the  analytical  calculus  from 
all  geometrical  bonds,  and  thus  to  have  established  analysis  as 
an  independent  science,  which  from  his  time  on  has  maintained 
an  unchallenged  leadership  in  the  field  of  mathematics. 

HANKEL,  H. 

Die  Entwickelung  der  Mathematik  in  den 
letzten  Jahrhunderten  (Tubingen,  1884),  P-  12. 

*  Riccardi's  Bibliografia  Euclidea  (Bologna,  1887),  lists  nearly  two 
thousand  editions. 


154  MEMORABILIA   MATHEMATICA 

967.  We  may  safely  say,  that  the  whole  form  of  modern 
mathematical  thinking  was  created  by  Euler.  It  is  only  with 
the  greatest  difficulty  that  one  is  able  to  follow  the  writings  of 
any  author  immediately  preceding  Euler,  because  it  was  not 
yet  known  how  to  let  the  formulas  speak  for  themselves.  This 
art  Euler  was  the  first  one  to  teach. — RUDIO,  F. 

Quoted  by  Ahrens  W.:  Scherz  und  Ernst  in  der 
Mathematik  (Leipzig,  1904),  P-  251- 

958.  The  general  knowledge  of  our  author  [Leonhard  Euler] 
was  more  extensive  than  could  well  be  expected,  in  one  who  had 
pursued,    with    such    unremitting    ardor,    mathematics    and 
astronomy  as  his  favorite  studies.    He  had  made  a  very  con- 
siderable progress  in  medical,  botanical,  and  chemical  science. 
What  was  still  more  extraordinary,  he  was  an  excellent  scholar, 
and  possessed  in  a  high  degree  what  is  generally  called  erudition. 
He  had  attentively  read  the  most  eminent  writers  of  ancient 
Rome;  the  civil  and  literary  history  of  all  ages  and  all  nations 
was  familiar  to  him;  and  foreigners,  who  were  only  acquainted 
with  his  works,  were  astonished  to  find  in  the  conversation  of  a 
man,  whose  long  life  seemed  solely  occupied  in  mathematical 
and  physical  researches  and  discoveries,  such  an  extensive  ac- 
quaintance with  the  most  interesting  branches  of  literature. 
In  this  respect,  no  doubt,  he  was  much  indebted  to  an  uncommon 
memory,  which  seemed  to  retain  every  idea  that  was  conveyed 
to  it,  either  from  reading  or  from  meditation. 

HUTTON,  CHARLES. 

Philosophical   and   Mathematical   Dictionary 
(London,  1815),  pp.  493-494- 

959.  Euler  could  repeat  the  Aeneid  from  the  beginning  to  the 
end,  and  he  could  even  tell  the  first  and  last  lines  hi  every  page 
of  the  edition  which  he  used.    In  one  of  his  works  there  is  a 
learned  memoir  on  a  question  hi  mechanics,  of  which,  as  he 
himself  informs  us,  a  verse  of  Aeneid  *  gave  him  the  first  idea. 

BREWSTER,  DAVID. 

Letters  of  Euler  (New  York,  1872},  Vol.  1, 
p.  24. 
*  The  line  referred  to  is : 

"The  anchor  drops,  the  rushing  keel  is  staid." 


PERSONS   AND   ANECDOTES  155 

960.  Most  of  his  [Euler's]  memoirs  are  contained  in  the  tran- 
sactions of  the  Academy  of  Sciences  at  St.  Petersburg,  and  in 
those  of  the  Academy  at  Berlin.    From  1728  to  1783  a  large 
portion  of  the  Petropolitan  transactions  were  filled  by  his 
writings.    He  had  engaged  to  furnish  the  Petersburg  Academy 
with  memoirs  in  sufficient  number  to  enrich  its  acts  for  twenty 
years — a  promise  more  than  fulfilled,  for  down  to  1818  [Euler 
died  in  1793]  the  volumes  usually  contained  one  or  more  papers 
of  his.     It  has  been  said  that  an  edition  of  Euler's  complete 
works  would  fill  16,000  quarto  pages. — CAJORI,  F. 

History  of  Mathematics  (New  York,  1897} , 
pp.  253-254. 

961.  Euler   who    could   have   been   called   almost   without 
metaphor,  and  certainly  without  hyperbole,  analysis  incarnate. 

ARAGO. 

Oeuvres,  t.  2  (1854),  P-  448. 

962.  Euler  calculated  without  any  apparent  effort,  just  as 
men  breathe,  as  eagles  sustain  themselves  in  the  air. — ARAGO. 

Oeuvres,  t.  2  (1854),  P-  133. 

963.  Two  of  his  [Euler's]  pupils  having  computed  to  the  17th 
term,  a  complicated  converging  series,  their  results  differed  one 
unit  in  the  fiftieth  cipher;  and  an  appeal  being  made  to  Euler, 
he  went  over  the  calculation  in  his  mind,  and  his  decision  was 
found  correct. — BREWSTER,  DAVID. 

Letters  of  Euler  (New  York,  1872),  Vol.  2, 
p.  22. 

964.  In  1735  the  solving  of  an  astronomical  problem,  pro- 
posed by  the  Academy,  for  which  several  eminent  mathemati- 
cians had  demanded  several  months'  time,  was  achieved  in 
three  days  by  Euler  with  aid  of  improved  methods  of  his 
own.  .  .  .     With  still  superior  methods  this  same  problem  was 
solved  by  the  illustrious  Gauss  in  one  hour. — CAJORI,  F. 

History  of  Mathematics  (New  York,  1897), 
p.  248. 


156  MEMORABILIA   MATHEMATICA 

965.  Euler's   Tentamen  novae  theorae  musicae  had  no  great 
success,  as  it  contained  too  much  geometry  for  musicians,  and 
too  much  music  for  geometers. — Fuss,  N. 

Quoted  by  Brewster:  Letters  of  Eider  (New  York, 
1872),  Vol.  1,  p.  26. 

966.  Euler  was  a  believer  in  God,  downright  and  straight- 
forward.    The  following  story  is  told  by  Thiebault,  in  his 
Souvenirs  de  vingt  ans  de  sijour  a  Berlin,  .  .  .  Thiebault  says 
that  he  has  no  personal  knowledge  of  the  truth  of  the  story, 
but  that  it  was  believed  throughout  the  whole  of  the  north  of 
Europe.     Diderot  paid  a  visit  to  the  Russian  Court  at  the 
invitation  of  the  Empress.    He  conversed  very  freely,  and  gave 
the  younger  members  of  the  Court  circle  a  good  deal  of  lively 
atheism.     The  Empress  was  much  amused,  but  some  of  her 
counsellors  suggested  that  it  might  be  desirable  to  check  these 
expositions  of  doctrine.     The  Empress  did  not  like  to  put  a 
direct  muzzle  on  her  guest's  tongue,  so  the  following  plot  was 
contrived.    Diderot  was  informed  that  a  learned  mathematician 
was  in  possession  of  an  algebraical  demonstration  of  the  ex  st- 
ence  of  God,  and  would  give  it  him  before  all  the  Court,  if  he 
desired  to  hear  it.    Diderot  gladly  consented :  though  the  name 
of  the  mathematician  is  not  given,  it  was  Euler.    He  advanced 
toward  Diderot,  and  said  gravely,  and  in  a  tone  of  perfect  con- 
viction: 

a  +  bn 

Monsieur, =  x,  done  Dieu  existe;  repondez! 

n 

Diderot,  to  whom  algebra  was  Hebrew,  was  embarrassed  and 
disconcerted;  while  peals  of  laughter  rose  on  all  sides.  He  asked 
permission  to  return  to  France  at  once,  which  was  granted. 

DE  MORGAN,  A. 

Budget  of  Paradoxes  (London,  1872),  p.  251. 

967.  Fermat  died  with  the  belief  that  he  had  found  a  long- 
sought-for  law  of  prime  numbers  in  the  formula  22"1  +  1  =  a 
prime,  but  he  admitted  that  he  was  unable  to  prove  it  rigor- 
ously. The  law  is  not  true,  as  was  pointed  out  by  Euler  in  the 
example  225  +  1  =  4,294,967,297  =  6,700,417  times  641.  The 
American  lightning  calculator  Zerah  Colburn,  when  a  boy, 


PERSONS  AND  ANECDOTES  157 

readily  found  the  factors  but  was  unable  to  explain  the  method 
by  which  he  made  his  marvellous  mental  computation. 

CAJORI,  F. 

History  of  Mathematics   (New   York,   1897), 

p.  180. 

968.  I  crave  the  liberty  to  conceal  my  name,  not  to  suppress 
it.     I  have  composed  the  letters  of  it  written  in  Latin  in  this 
sentence — 

In  Mathesi  a  sole  fundes.* — FLAMSTEED,  J. 

Macclesfield:  Correspondence  of  Scientific  Men 
(Oxford,  1841),  Vol.  2,  p.  90. 

969.  To  the  Memory  of  Fourier 

Fourier!  with  solemn  and  profound  delight, 

Joy  born  of  awe,  but  kindling  momently 

To  an  intense  and  thrilling  ecstacy, , 

I  gaze  upon  thy  glory  and  grow  bright: 

As  if  irradiate  with  beholden  light; 

As  if  the  immortal  that  remains  of  thee 

Attuned  me  to  thy  spirit's  harmony, 

Breathing  serene  resolve  and  tranquil  might. 

Revealed  appear  thy  silent  thoughts  of  youth, 

As  if  to  consciousness,  and  all  that  view 

Prophetic,  of  the  heritage  of  truth 

To  thy  majestic  years  of  manhood  due: 

Darkness  and  error  fleeing  far  away, 

And  the  pure  mind  enthroned  in  perfect  day. 

HAMILTON,  W.  R. 

Graves'  Life  of  W.  R.  Hamilton,  (New  York, 

1882),  Vol.  l,p.596. 

970.  Astronomy  and  Pure  Mathematics  are  the  magnetic 
poles  toward  which  the  compass  of  my  mind  ever  turns. 

GAUSS  TO  BOLYAI. 
Briefwechsel  (Schmidt-Stakel) ,  (1899),  p.  55. 

971.  [Gauss  calculated  the  elements  of  the  planet  Ceres]  and 
his  analysis  proved  him  to  be  the  first  of  theoretical  astronomers 
no  less  than  the  greatest  of  "arithmeticians." — BALL,  W.  W.  R. 

History  of  Mathematics  (London,  1901),  p.  458. 

*  Johannes  Flamsteedius. 


158  MEMORABILIA   MATHEMATICA 

972.  The  mathematical  giant  [Gauss],  who  from  his  lofty 
heights  embraces  in  one  view  the  stars  and  the  abysses  .  .  . 

BOLYAI,  W. 

Kurzer  Grundriss  eines  Versuchs  (Maros  Vas- 
arhely,  1851),  p.  44- 

973.  Almost  everything,  which  the  mathematics  of  our  cen- 
tury has  brought  forth  in  the  way  of  original  scientific  ideas, 

attaches  to  the  name  of  Gauss. — KRONECKER,  L. 

Zahlentheorie,  Teil  1  (Leipzig,  1901),  p.  48. 

974.  I  am  giving  this  whiter  two  courses  of  lectures  to  three 
students,  of  which  one  is  only  moderately  prepared,  the  other 
less  than  moderately,  and  the  third  lacks  both  preparation  and 
ability.     Such  are  the  onera  of  a  mathematical  profession. 

GAUSS  TO  BESBEL,  1810. 
Gauss-Bessel  Briefwechsel  (1880),  p.  107. 

975.  Gauss  once  said  "Mathematics  is  the  queen  of  the 
sciences  and  number-theory  the  queen  of  mathematics."     If 
this  be  true  we  may  add  that  the  Disquisitiones  is  the  Magna 
Charta  of  number-theory.    The  advantage  which  science  gained 
by  Gauss '  long-lingering  method  of  publication  is  this :  What  he 
put  into  print  is  as  true  and  important  today  as  when  first  pub- 
lished; his  publications  are  statutes,  superior  to  other  human 
statutes  in  this,  that  nowhere  and  never  has  a  single  error  been 
detected  hi   them.     This  justifies  and  makes  intelligible  the 
pride  with  which  Gauss  said  in  the  evening  of  his  life  of  the  first 
larger  work  of  his  youth:   "The  Disquisitiones  arithmeticae 
belong  to  history." — CANTOR,  M. 

Allgemeine  Deutsche  Biographie,  Bd.  8  (1878), 
p.  435. 

976.  Here  I  am  at  the  limit  which  God  and  nature  has  as- 
signed to  my  individuality.    I  am  compelled  to  depend  upon 
word,  language  and  image  in  the  most  precise  sense,  and  am 
wholly  unable  to  operate  in  any  manner  whatever  with  symbols 
and  numbers  which  are  easily  intelligible  to  the  most  highly 
gifted  minds. — GOETHE. 

Letter  to  Naumann   (1826);   Vogel:  Goethe's 
Selbstzeugnisse  (Leipzig,  1903),  p.  56. 


PERSONS  AND  ANECDOTES  159 

977.  Dirichlet  was  not  satisfied  to  study  Gauss'  "  Disquisi- 
tiones  arithmeticae "   once  or    several   times,  but  continued 
throughout  life  to  keep  in  close  touch  with  the  wealth  of  deep 
mathematical  thoughts  which  it  contains  by  perusing  it  again 
and  again.    For  this  reason  the  book  was  never  placed  on  the  shelf 
but  had  an  abiding  place  on  the  table  at  which  he  worked.  .  .  . 
Dirichlet  was  the  first  one,  who  not  only  fully  understood  this 
work,  but  made  it  also  accessible  to  others. — KUMMER,  E.  E. 

Dirichlet:  Werke,  Bd.  2,  p.  315. 

978.  [The  famous  attack  of  Sir  William  Hamilton  on  the 
tendency  of  mathematical  studies]  affords  the  most  express 
evidence  of  those  fatal  lacunae  in  the  circle  of  his  knowledge, 
which  unfitted  him  for  taking  a  comprehensive  or  even  an 
accurate  view  of  the  processes  of  the  human  mind  in  the  es- 
tablishment of  truth.     If  there  is  any  pre-requisite  which  all 
must  see  to  be  indispensable  in  one  who  attempts  to  give  laws  to 
the  human  intellect,  it  is  a  thorough  acquaintance  with  the  modes 
by  which  human  intellect  has  proceeded,  in  the  case  where,  by 
universal   acknowledgment,    grounded   on   subsequent   direct 
verification,  it  has  succeeded  in  ascertaining  the  greatest  num- 
ber of  important  and  recondite  truths.    This  requisite  Sir  W. 
Hamilton  had  not,  in  any  tolerable  degree,  fulfilled.    Even  of 
pure  mathematics  he  apparently  knew  little  but  the  rudiments. 
Of  mathematics  as  applied  to  investigating  the  laws  of  physi- 
cal nature;  of  the  mode  in  which  the  properties  of  number, 
extension,  and  figure,  are  made  instrumental  to  the  ascertain- 
ment of  truths  other  than  arithmetical  or  geometrical — it  is  too 
much  to  say  that  he  had  even  a  superficial  knowledge:  there  is 
not  a  line  in  his  works  which  shows  him  to  have  had  any  knowl- 
edge at  all. — MILL,  J.  S. 

Examination  of  Sir  William  Hamilton's  Philos- 
ophy (London,  1878},  p.  607. 

979.  Helmholtz — the  physiologist  who  learned  physics  for  the 
sake  of  his  physiology,  and  mathematics  for  the  sake  of  his 
physics,  and  is  now  in  the  first  rank  of  all  three. 

CLIFFORD,  W.  K. 

.At'ms  and  Instruments  of  Scientific  Thought; 
Lectures  and  Essays,  Vol.  1  (London,  1901), 
p.  165. 


160  MEMORABILIA   MATHEMATICA 

980.  It  is  said  of  Jacobi,  that  he  attracted  the  particular 
attention  and  friendship  of  Bockh,  the  director  of  the  philologi- 
cal seminary  at  Berlin,  by  the  great  talent  he  displayed  for 
philology,  and  only  at  the  end  of  two  years'  study  at  the  Univer- 
sity, and  after  a  severe  mental  struggle,  was  able  to  make  his 
final  choice  in  favor  of  mathematics. — SYLVESTER,  J.  J. 

Collected  Mathematical  Papers,  Vol.  2  (Cam- 
bridge, 1908),  p.  651. 

981.  When  Dr.  Johnson  felt,  or  fancied  he  felt,  his  fancy  dis- 
ordered, his  constant  recurrence  was  to  the  study  of  arithmetic. 

BOSWELL,  J. 

Life  of  Johnson   (Harper1  s  Edition,   1871), 
Vol.  2,  p.  264. 

982.  Endowed  with  two  qualities,  which  seemed  incompatible 
with  each  other,  a  volcanic  imagination  and  a  pertinacity  of 
intellect  which  the  most  tedious  numerical  calculations  could 
not  daunt,  Kepler  conjectured  that  the  movements  of  the 
celestial  bodies  must  be  connected  together  by  simple  laws, 
or,  to  use  his  own  expression,  by  harmonic  laws.    These  laws  he 
undertook  to  discover.    A  thousand  fruitless  attempts,  errors  of 
calculation  inseparable  from  a  colossal  undertaking,  did  not  pre- 
vent him  a  single  instant  from  advancing  resolutely  toward  the 
goal  of  which  he  imagined  he  had  obtained  a  glimpse.    Twenty- 
two  years  were  employed  by  him  in  this  investigation,  and 
still  he  was  not  weary  of  it!  What,  hi  reality,  are  twenty-two 
years  of  labor  to  him  who  is  about  to  become  the  legislator  of 
worlds;  who  shall  inscribe  his  name  in  ineffaceable  characters 
upon  the  frontispiece  of  an  immortal  code;  who  shall  be  able  to 
exclaim  in  dithyrambic  language,  and  without  incurring  the 
reproach  of  anyone,    "The  die  is  cast;  I  have  written  my  book; 
it  will  be  read  either  in  the  present  age  or  by  posterity,  it  matters 
not  which;  it  may  well  await  a  reader,  since  God  has  waited  six 
thousand  years   for  an   interpreter  of  his  words." — ARAGO. 

Eulogy  on  Laplace:   [Baden  Powell]  Smith- 
sonian Report,  1874,  P-  132. 

983.  The  great  masters  of  modern  analysis  are  Lagrange, 
Laplace,  and  Gauss,  who  were  contemporaries.    It  is  interesting 


PERSONS   AND   ANECDOTES  161 

to  note  the  marked  contrast  in  their  styles.  Lagrange  is  perfect 
both  in  form  and  matter,  he  is  careful  to  explain  his  procedure, 
and  though  his  arguments  are  general  they  are  easy  to  follow. 
Laplace  on  the  other  hand  explains  nothing,  is  indifferent  to 
style,  and,  if  satisfied  that  his  results  are  correct,  is  content  to 
leave  them  either  with  no  proof  or  with  a  faulty  one.  Gauss 
is  as  exact  and  elegant  as  Lagrange,  but  even  more  difficult  to 
follow  than  Laplace,  for  he  removes  every  trace  of  the  analysis 
by  which  he  reached  his  results,  and  studies  to  give  a  proof  which 
while  rigorous  shall  be  as  concise  and  synthetical  as  possible. 

BALL,  W.  W.  R. 

History  of  Mathematics  (London,  1901),  p.  463. 

984.  Lagrange,  in  one  of  the  later  years  of  his  life,  imagined 
that  he  had  overcome  the  difficulty  [of  the  parallel  axiom]. 
He  went  so  far  as  to  write  a  paper,  which  he  took  with  him  to  the 
Institute,  and  began  to  read  it.     But  in  the  first  paragraph 
something  struck  him  which  he  had  not  observed :  he  muttered 
II  faut  que  fy  songe  encore,  and  put  the  paper  in  his  pocket. 

DE  MORGAN,  A. 
Budget  of  Paradoxes  (London,  1872),  p.  178. 

985.  I  never  come  across  one  of  Laplace's  "  Thus  it  plainly 
appears  "  without  feeling  sure  that  I  have  hours  of  hard  work 
before  me  to  fill  up  the  chasm  and  find  out  and  show  how  it 
plainly  appears. — BOWDITCH,  N. 

Quoted  by  Cajori:  Teaching  and  History  of 
Mathematics  in  the  U.  S.  (Washington,  1896), 
p.  104. 

986.  Biot,  who  assisted  Laplace  in  revising  it  [The  Me"cani- 
que  Celeste]  for  the  press,  says  that  Laplace  himself  was  fre- 
quently unable  to  recover  the  details  in  the  chain  of  reasoning, 
and  if  satisfied  that  the  conclusions  were  correct,  he  was  con- 
tent to  insert  the  constantly  recurring  formula,  "  //  est  dise  a 
voir."— BALL,  W.  W.  R. 

History  of  Mathematics  (London,  1901),  p  427. 

987.  It  would  be  difficult  to  name  a  man  more  remarkable 
for  the  greatness  and  the  universality  of  his  intellectual  powers 
than  Leibnitz. — MILL,  J.  S. 

System  of  Logic,  Bk.  2,  chap.  5,  sect.  6. 


162  MEMORABILIA   MATHEMATICA 

988.  The  influence  of  his  [Leibnitz's]  genius  in  forming  that 
peculiar  taste  both  hi  pure  and  in  mixed  mathematics  which  has 
prevailed  in  France,  as  well  as  in  Germany,  for  a  century  past, 
will  be  found,  upon  examination,  to  have  been  incomparably 
greater  than  that  of  any  other  individual. — STEWART,  DUGALD. 

Philosophy  of  the  Human  Mind,  Part  8,  chap. 
1,  sect.  8. 

989.  Leibnitz's  discoveries  lay  in  the  direction  in  which  all 
modern  progress  in  science  lies,  in  establishing  order,  symmetry, 
and  harmony,  i.  e.,  comprehensiveness  and  perspicuity, — rather 
than  hi  dealing  with  single  problems,  hi  the  solution  of  which 
followers  soon  attained  greater  dexterity  than  himself. 

MERZ,  J    T. 
Leibnitz,  Chap.  6. 

990.  It  was  his  [Leibnitz's]  love  of  method  and  order,  and  the 
conviction  that  such  order  and  harmony  existed  hi  the  real 
world,  and  that  our  success  in  understanding  it  depended  upon 
the  degree  and  order  which  we  could  attain  in  our  own  thoughts, 
that  originally  was  probably  nothing  more  than  a  habit  which 
by  degrees  grew  into  a  formal  rule.*    This  habit  was  acquired 
by  early  occupation  with  legal  and  mathematical  questions. 
We  have  seen  how  the  theory  of  combinations  and  arrangements 
of  elements  had  a  special  interest  for  him.    We  also  saw  how 
mathematical  calculations  served  him  as  a  type  and  model  of 
clear  and  orderly  reasoning,  and  how  he  tried  to  introduce 
method  and  system  into  logical  discussions,  by  reducing  to  a 
small  number  of  terms  the  multitude  of  compound  notions  he 
had  to  deal  with.     This  tendency  increased  in  strength,  and 
even  in  those  early  years  he  elaborated  the  idea  of  a  general 
arithmetic,  with  a  universal  language  of  symbols,  or  a  character- 
istic which  would  be  applicable  to  all  reasoning  processes,  and 
reduce   philosophical   investigations   to   that   simplicity   and 
certainty  which  the  use  of  algebraic  symbols  had  introduced 
into  mathematics. 

A  mental  attitude  such  as  this  is  always  highly  favorable  for 
mathematical  as  well  as  for  philosophical  investigations.    Wher- 

*  This  sentence  has  been  reworded  for  the  purpose  of  this  quotation. 


PERSONS  AND  ANECDOTES  163 

ever  progress  depends  upon  precision  and  clearness  of  thought, 
and  wherever  such  can  be  gained  by  reducing  a  variety  of  in- 
vestigations to  a  general  method,  by  bringing  a  multitude  of 
notions  under  a  common  term  or  symbol,  it  proves  inestimable. 
It  necessarily  imports  the  special  qualities  of  number — viz., 
their  continuity,  infinity  and  infinite  divisibility — like  mathe- 
matical quantities — and  destroys  the  notion  that  irreconcilable 
contrasts  exist  in  nature,  or  gaps  which  cannot  be  bridged  over. 
Thus,  in  his  letter  to  Arnaud,  Leibnitz  expresses  it  as  his  opinion 
that  geometry,  or  the  philosophy  of  space,  forms  a  step  to  the 
philosophy  of  motion — i.  e.,  of  corporeal  things — and  the 
philosophy  of  motion  a  step  to  the  philosophy  of  mind. 

MERZ,  J.  T. 
Leibnitz    (Philadelphia,),  pp.  44~4-5- 

991.  Leibnitz  believed  he  saw  the  image  of  creation  in  his 
binary  arithmetic  in  which  he  employed  only  two  characters, 
unity  and  zero.    Since  God  may  be  represented  by  unity,  and 
nothing  by  zero,  he  imagined  that  the  Supreme  Being  might 
have  drawn  all  things  from  nothing,  just  as  in  the  binary  arith- 
metic all  numbers  are  expressed  by  unity  with  zero.    This  idea 
was  so  pleasing  to  Leibnitz,  that  he  communicated  it  to  the 
Jesuit  Grimaldi,  President  of  the  Mathematical  Board  of  China, 
with  the  hope  that  this  emblem  of  the  creation  might  convert 
to  Christianity  the  reigning  emperor  who  was  particularly  at- 
tached to  the  sciences. — LAPLACE. 

Essai    Philosophique    sur    les    Probabilites; 
Oeuvres  (Paris,  1896),  t.  7,  p.  119. 

992.  Sophus  Lie,  great  comparative  anatomist  of  geometric 
theories. — KEYSER,  C.  J. 

Lectures  on  Science,  Philosophy  and  Art  (New 
York,  1908),  p.  81. 

993.  It  has  been  the  final  aim  of  Lie  from  the  beginning  to 
make  progress  in  the  theory  of  differential  equations;  as  sub- 
sidiary to  this  may  be  regarded  both  his  geometrical  develop- 
ments and  the  theory  of  continuous  groups. — KLEIN,  F. 

Lectures  on  Mathematics  (New  York,  1911), 
p.  24. 


164  MEMORABILIA   MATHEMATICA 

994.  To  fully  understand  the  mathematical  genius  of  Sophus 
Lie,  one  must  not  turn  to  books  recently  published  by  him  in 
collaboration  with  Dr.  Engel,  but  to  his  earlier  memoirs,  written 
during  the  first  years  of  his  scientific  career.    There  Lie  shows 
himself  the  true  geometer  that  he  is,  while  in  his  later  publica- 
tions, finding  that  he  was  but  imperfectly  understood  by  the 
mathematicians  accustomed  to  the  analytic  point  of  view,  he 
adopted  a  very  general  analytic  form  of  treatment  that  is  not 
always  easy  to  follow. — KLEIN,  F. 

Lectures  on  Mathematics  (New  York,  19 11),  p.  9. 

995.  It  is  said  that  the  composing  of  the  Lilawati  was  oc- 
casioned by  the  following  circumstance.     Lilawati  was  the 
name  of  the  author's  [Bhascara]  daughter,  concerning  whom  it 
appeared,  from  the  qualities  of  the  ascendant  at  her  birth,  that 
she  was  destined  to  pass  her  life  unmarried,  and  to  remain 
without  children.     The  father  ascertained  a  lucky  hour  for 
contracting  her  in  marriage,  that  she  might  be  firmly  con- 
nected and  have  children.    It  is  said  that  when  that  hour  ap- 
proached, he  brought  his  daughter  and  his  intended  son  near 
him.    He  left  the  hour  cup  on  the  vessel  of  water  and  kept  in 
attendance  a  time-knowing  astrologer,  in  order  that  when  the 
cup  should  subside  in  the  water,  those  two  precious  jewels 
should  be  united.    But,  as  the  intended  arrangement  was  not 
according  to  destiny,  it  happened  that  the  girl,  from  a  curiosity 
natural  to  children,  looked  into  the  cup,  to  observe  the  water 
coming  in  at  the  hole,  when  by  chance  a  pearl  separated  from 
her  bridal  dress,  fell  into  the  cup,  and,  rolling  down  to  the  hole, 
stopped  the  influx  of  water.    So  the  astrologer  waited  in  expecta- 
tion of  the  promised  hour.    When  the  operation  of  the  cup  had 
thus  been  delayed  beyond  all  moderate  time,  the  father  was  in 
consternation,  and  examining,  he  found  that  a  small  pearl  had 
stopped  the  course  of  the  water,  and  that  the  long-expected 
hour  was  passed.    In  short,  the  father,  thus  disappointed,  said 
to  his  unfortunate  daughter,  I  will  write  a  book  of  your  name, 
which  shall  remain  to  the  latest  times — for  a  good  name  is  a 
second  life,  and  the  ground-work  of  eternal  existence. — FIZI. 

Preface  to  the  Lilawati.  Quoted  by  A.  Hutton: 
A  Philosophical  and  Mathematical  Dictionary, 
Article  "Algebra"  (London,  1815). 


PERSONS  AND  ANECDOTES  165 

996.  Is  there  anyone  whose  name  cannot  be  twisted  into 
either  praise  or  satire?    I  have  had  given  to  me, 
Thomas  Babington  Macaulay 
Mouths  big:  a  Cantab  anomaly. 

DE  MORGAN,  A. 
Budget  of  Paradoxes  (London,  1872),  p.  83. 


CHAPTER  X 

PERSONS  AND   ANECDOTES 

(N-Z) 

1001.  When  he  had  a  tew  moments  for  diversion,  he  [Napo- 
leon] not  unfrequently  employed  them  over  a  book  of  logarithms, 
in  which  he  always  found  recreation. — ABBOTT,  J.  S.  C. 

Napoleon  Bonaparte  (New  York,  1904),  Vol.  1, 
chap,  10. 

1002.  The  name  of  Sir  Isaac  Newton  has  by  general  consent 
been  placed  at  the  head  of  those  great  men  who  have  been  the 
ornaments  of  their  species.  .  .  .  The  philosopher  [Laplace], 
indeed,  to  whom  posterity  will  probably  assign  a  place  next  to 
Newton,  has  characterized  the  Principia  as  pre-eminent  above 

all  the  productions  of  human  intellect. — BREWSTER,  D. 

Life  of  Sir  Isaac  Newton  (London,  1831),  pp.  1, 2. 

1003.  Newton  and  Laplace  need  myriads  of  ages  and  thick- 
strewn  celestial  areas.    One  may  say  a  gravitating  solar  system 
is  already  prophesied  hi  the  nature  of  Newton's  mind. 

EMERSON. 
Essay  on  History. 

1004.  The  law  of  gravitation  is  indisputably  and  incompar- 
ably the  greatest  scientific  discovery  ever  made,  whether  we 
look  at  the  advance  which  it  involved,  the  extent  of  truth  dis- 
closed, or  the  fundamental  and  satisfactory  nature  of  this 

truth. — WHEWELL,  W. 

History  of  the  Inductive  Sciences,  Bk.  7,  chap.  2, 
sect.  5. 

1005.  Newton's  theory  is  the  circle  of  generalization  which 
includes  all  the  others  [as  Kepler's  laws,  Ptolemy's  theory, 
etc.]; — the  highest  point  of  the  inductive  ascent; — the  catas- 
trophe of  the  philosophic  drama  to  which  Plato  had  prologized; — 

166 


PERSONS   AND   ANECDOTES  167 

the  point  to  which  men's  minds  had  been  journeying  for  two 
thousand  years. — WHEWELL,  W. 

History  of  the  Inductive  Sciences,  Bk.  7,  chap.  2, 
sect.  5. 

1006.  The  efforts  of  the  great  philosopher  [Newton]  were 
always  superhuman;  the  questions  which  he  did  not  solve  were 

incapable  of  solution  in  his  time. — ARAGO. 

Eulogy  on  Laplace,   [Baden  Powell]  Smith- 
sonian Report,  1874,  p.  1S3. 

1007.  Nature  and  Nature's  laws  lay  hid  in  night: 
God  said,  "  Let  Newton  be!  "  and  all  was  light. 

POPE,  A. 

Epitaph  intended  for  Sir  Isaac  Newton. 

1008.  There  Priest  of  Nature!  dost  thou  shine, 

Newton !  a  King  among  the  Kings  divine. — SOUTHEY. 
Translation  of  a  Greek  Ode  on  Astronomy. 

1009.  O'er  Nature's  laws  God  cast  the  veil  of  night, 
Out-blaz'd  a  Newton's  soul — and  all  was  light. 

HILL,  AARON. 
On  Sir  Isaac  Newton. 

1010.  Taking  mathematics  from  the  beginning  of  the  world 
to  the  time  when  Newton  lived,  what  he  had  done  was  much  the 
better  half. — LEIBNITZ. 

Quoted  by  F.   R.   Moulton:  Introduction  to 
Astronomy  (New  York,  1906},  p.  199. 

1011.  Newton  was  the  greatest  genius  that  ever  existed,  and 
the  most  fortunate,  for  we  cannot  find  more  than  once  a  system 
of  the  world  to  establish. — LAGRANGE. 

Quoted  by  F.   R.  Moulton:  Introduction  to 
Astronomy  (New  York,  1906),  p.  199. 

1012.  A  monument  to  Newton!  a  monument  to  Shakespeare! 
Look  up  to  Heaven — look  into  the  Human  Heart.     Till  the 
planets  and  the  passions — the  affections  and  the  fixed  stars  are 
extinguished — their  names  cannot  die. — WILSON,  JOHN. 

Noctes  Ambrosianae. 


168  MEMORABILIA   MATHEMATICA 

1013.  Such  men  as  Newton  and  Linnaeus  are  incidental,  but 
august,  teachers  of  religion. — WILSON,  JOHN. 

Essays:  Education  of  the  People. 

1014.  Sir    Isaac   Newton,    the   supreme   representative   of 
Anglo-Saxon  genius. — ELLIS,  HAVELOCK. 

Study  of  British  Genius  (London,  1904),  P-  49. 

1015.  Throughout  his  life  Newton  must  have  devoted  at 
least  as  much  attention  to  chemistry  and  theology  as  to  mathe- 
matics. .  .  . — BALL,  W.  W.  R. 

History  of  Mathematics  (London,  1901),  p.  385. 

1016.  There  was  a  tune  when  he  [Newton]  was  possessed  with 
the  old  fooleries  of  astrology;  and  another  when  he  was  so  far 
gone  in  those  of  chemistry,  as  to  be  upon  the  hunt  after  the 
philosopher's  stone. — REV.  J.  SPENCE. 

Anecdotes,    Observations,    and   Characters    of 
Books  and  Men  (London,  1858),  p.  54- 

1017.  For  several  years  this  great  man  [Newton]  was  in- 
tensely occupied  in  endeavoring  to  discover  a  way  of  changing 
the  base  metals  into  gold.  .  .  .  There  were  periods  when  his 
furnace  fires  were  not  allowed  to  go  out  for  six  weeks;  he  and 
his  secretary  sitting  up  alternate  nights  to  replenish  them. 

PARTON,  JAMES. 
Sir  Isaac  Newton. 

1018.  On  the  day  of  Cromwell's  death,  when  Newton  was 
sixteen,  a  great  storm  raged  all  over  England.    He  used  to  say, 
in  his  old  age,  that  on  that  day  he  made  his  first  purely  scientific 
experiment.    To  ascertain  the  force  of  the  wind,  he  first  jumped 
with  the  wind  and  then  against  it;  and,  by  comparing  these 
distances  with  the  extent  of  his  own  jump  on  a  calm  day,  he  was 
enabled  to  compute  the  force  of  the  storm.    When  the  wind 
blew  thereafter,  he  used  to  say  it  was  so  many  feet  strong. 

PARTON,  JAMES. 
Sir  Isaac  Newton. 

1019.  Newton  lectured  now  and  then  to  the  few  students  who 
chose  to  hear  him;  and  it  is  recorded  that  very  frequently  he 


PERSONS   AND    ANECDOTES  169 

came  to  the  lecture-room  and  found  it  empty.  On  such  occa- 
sions he  would  remain  fifteen  minutes,  and  then,  if  no  one  came, 
return  to  his  apartments. — PARTON,  JAMES. 

Sir  Isaac  Newton. 

1020.  Sir  Isaac  Newton,  though  so  deep  in  algebra  and 
fluxions,  could  not  readily  make  up  a  common  account:  and, 
when  he  was  Master  of  the  Mint,  used  to  get  somebody  else  to 
make  up  his  accounts  for  him. — REV.  J.  SPENCE. 

Anecdotes,    Observations,    and    Characters    of 
Books  and  Men  (London,  1858},  p.  132. 

1021.  We  have  one  of  his  [Newton's]  college  memorandum- 
books,  which  is  highly  interesting.    The  following  are  some  of 
the  entries:  "Drills,  gravers,  a  hone,  a  hammer,  and  a  man- 
dril, 5s.;"  "a  magnet,  16s.;"  "compasses,  2s.;"  "glass  bubbles, 
4s.;"  "at  the  tavern  several  other  times,  £1;"  "spent  on  my 
cousin,  12s.;"  "on  other  acquaintances,  10s.;"  "Philosophical 
Intelligences,  9s.  6d.;"  "lost  at  cards  twice,  15s.;"  "at  the 
tavern  twice,  3s.  6d.;"  "to  three  prisms,  £3;"  "four  ounces  of 
putty,  Is.  4d.;"  "Bacon's  Miscellanies,  Is.  6d.;"  "a  bible  bind- 
ing, 3s.;"  "for  oranges  to  my  sister,  4s.  2d.;"  "for  aquafortis, 
sublimate,  oyle  pink,  fine  silver,  antimony,  vinegar,  spirit  of 
wine,  white  lead,  salt  of  tartar,  £2;"  "Theatrum  chemicum, 
£1  8s." — PARTON,  JAMES. 

Sir  Isaac  Newton. 

1022.  On  one  occasion,  when  he  was  giving  a  dinner  to  some 
friends  at  the  university,  he  left  the  table  to  get  them  a  bottle 
of  wine;  but,  on  his  way  to  the  cellar,  he  fell  into  reflection,  for- 
got his  errand  and  his  company,  went  to  his  chamber,  put  on  his 
surplice,  and  proceeded  to  the  chapel.    Sometimes  he  would  go 
into  the  street  half  dressed,  and  on  discovering  his  condition, 
run  back  in  great  haste,  much  abashed.    Often,  while  strolling 
in  his  garden,  he  would  suddenly  stop,  and  then  run  rapidly  to 
his  room,  and  begin  to  write,  standing,  on  the  first  piece  of  paper 
that  presented  itself.    Intending  to  dine  in  the  public  hall,  he 
would  go  out  in  a  brown  study,  take  the  wrong  turn,  walk  a 
while,  and  then  return  to  his  room,  having  totally  forgotten  the 
dinner.    Once  having  dismounted  from  his  horse  to  lead  him 


170  MEMORABILIA    MATHEMATICA 

up  a  hill,  the  horse  slipped  his  head  out  of  the  bridle;  but  Newton, 
oblivious,  never  discovered  it  till,  on  reaching  a  tollgate  at  the 
top  of  the  hill,  he  turned  to  remount  and  perceived  that  the 
bridle  which  he  held  in  his  hand  had  no  horse  attached  to  it. 
His  secretary  records  that  his  forgetfulness  of  his  dinner  was  an 
excellent  thing  for  his  old  housekeeper,  who  "sometimes  found 
both  dinner  and  supper  scarcely  tasted  of,  which  the  old  woman 
has  very  pleasantly  and  mumpingly  gone  away  with."  On 
getting  out  of  bed  hi  the  morning,  he  has  been  discovered  to 
sit  on  his  bedside  for  hours  without  dressing  himself,  utterly 
absorbed  in  thought. — PARTON,  JAMES. 

Sir  Isaac  Newton. 

1023.  I  don't  know  what  I  may  seem  to  the  world,  but,  as  to 
myself,  I  seem  to  have  been  only  as  a  boy  playing  on  the  sea- 
shore, and  diverting  myself  in  now  and  then  finding  a  smoother 
pebble  or  a  prettier  shell  than  ordinary,  whilst  the  great  ocean 
of  truth  lay  all  undiscovered  before  me. — NEWTON,  I. 

Quoted  by  Rev.  J.  Spence:  Anecdotes,  Observa- 
tions, and  Characters  of  Books  and  Men  (Lon- 
don, 1858),  p.  40. 

1024.  If  I  have  seen  farther  than  Descartes,  it  is  by  standing 
on  the  shoulders  of  giants. — NEWTON,  I. 

Quoted  by  James  Parton:  Sir  Isaac  Newton. 

1025.  Newton  could  not  admit  that  there  was  any  difference 
between  him  and  other  men,  except  in  the  possession  of  such 
habits   as  ...  perseverance   and   vigilance.     When   he  was 
asked  how  he  made  his  discoveries,  he  answered,  "by  always 
thinking  about  them;"  and  at  another  time  he  declared  that 
if  he  had  done  anything,  it  was  due  to  nothing  but  industry  and 
patient  thought:  "I  keep  the  subject  of  my  inquiry  constantly 
before  me,  and  wait  till  the  first  dawning  opens  gradually,  by 

little  and  little,  into  a  full  and  clear  light."  WHEWELL,  W. 

History  of  the  Inductive  Sciences,  Bk.  7,  chap.  2, 
sect.  5. 

1026.  Newton  took  no  exercise,  indulged  hi  no  amusements, 
and  worked  incessantly,  often  spending  eighteen  or  nineteen 
hours  out  of  the  twenty-four  in  writing. — BALL,  W.  W.  R. 

History  of  Mathematics  (London,  1901),  p.  858. 


PERSONS  AND  ANECDOTES  171 

1027.  Foreshadowings  of  the  principles  and  even  of  the 
language  of  [the  infinitesimal]  calculus  can  be  found  in  the  writ- 
ings of  Napier,  Kepler,  Cavalieri,  Pascal,  Fermat,  Wallis,  and 
Barrow.    It  was  Newton's  good  luck  to  come  at  a  time  when 
everything  was  ripe  for  the  discovery,  and  his  ability  enabled 
him  to  construct  almost  at  once  a  complete  calculus. 

BALL,  W.  W.  R. 
History  of  Mathematics  (London,  1901),  p.  856. 

1028.  Kepler's  suggestion  of  gravitation  with  the  inverse 
distance,  and  Bouillaud's  proposed  substitution  of  the  inverse 
square  of  the  distance,  are  things  which  Newton  knew  better 
than  his  modern  readers.    I  have  discovered  two  anagrams  on 
his  name,  which  are  quite  conclusive:  the  notion  of  gravitation 
was  not  new;  but  Newton  went  on. — DE  MORGAN,  A. 

Budget  of  Paradoxes  (London,  1872},  p.  82. 

1029.  For  other  great  mathematicians  or  philosophers,  he 
[Gauss]  used  the  epithets  magnus,  or  clarus,  or  clarissimus;  for 
Newton  alone  he  kept  the  prefix  summus. — BALL,  W.  W.  R. 

History  of  Mathematics  (London,  1901),  p.  362. 

1030.  To  know  him  [Sylvester]  was  to  know  one  of  the  historic 
figures  of  all  time,  one  of  the  immortals;  and  when  he  was  really 
moved  to  speak,  his  eloquence  equalled  his  genius. 

HALSTED,  G.  B. 

F.  Cajori's  Teaching  and  History  of  Mathe- 
matics in  the  U.  S.  (Washington,  1890),  p.  265. 

1031.  Professor  Sylvester's  first  high  class  at  the  new  univer- 
sity Johns  Hopkins  consisted  of  only  one  student,  G.  B.  Hal- 
sted,  who  had  persisted  in  urging  Sylvester  to  lecture  on  the 
modern  algebra.     The  attempt  to  lecture  on  this  subject  led 
him  into  new  investigations  in  quantics. — CAJORI,  F. 

Teaching  and  History  of  Mathematics  in  the 
U.  S.  (Washington,  1890),  p.  26^. 

1032.  But  for  the  persistence  of  a  student  of  this  university  in 
urging  upon  me  his  desire  to  study  with  me  the  modern  algebra 
I  should  never  have  been  led  into  this  investigation;  and  the 


172  MEMORABILIA    MATHEMATICA 

new  facts  and  principles  which  I  have  discovered  in  regard  to 
it  (important  facts,  I  believe),  would,  so  far  as  I  am  concerned, 
have  remained  still  hidden  in  the  womb  of  time.  In  vain  I 
represented  to  this  inquisitive  student  that  he  would  do  better 
to  take  up  some  other  subject  lying  less  off  the  beaten  track  of 
study,  such  as  the  higher  parts  of  the  calculus  or  elliptic  func- 
tions, or  the  theory  of  substitutions,  or  I  wot  not  what  besides. 
He  stuck  with  perfect  respectfulness,  but  with  invincible  per- 
tinacity, to  his  point.  He  would  have  the  new  algebra  (Heaven 
knows  where  he  had  heard  about  it,  for  it  is  almost  unknown  in 
this  continent),  that  or  nothing.  I  was  obliged  to  yield,  and 
what  was  the  consequence?  In  trying  to  throw  light  upon  an 
obscure  explanation  in  our  text-book,  my  brain  took  fire,  I 
plunged  with  re-quickened  zeal  into  a  subject  which  I  had  for 
years  abandoned,  and  found  food  for  thoughts  which  have 
engaged  my  attention  for  a  considerable  tune  past,  and  will 
probably  occupy  all  my  powers  of  contemplation  advantageously 
for  several  months  to  come. — SYLVESTER,  J.  J. 

Johns  Hopkins  Commemoration  Day  Ad- 
dress; Collected  Mathematical  Papers,  Vol.  3, 
p.  76. 

1033.  Sylvester  was  incapable  of  reading  mathematics  in  a 
purely  receptive  way.  Apparently  a  subject  either  fired  in  his 
brain  a  train  of  active  and  restless  thought,  or  it  would  not  re- 
tain his  attention  at  all.  To  a  man  of  such  a  temperament,  it 
would  have  been  peculiarly  helpful  to  live  in  an  atmosphere  in 
which  his  human  associations  would  have  supplied  the  stimulus 
which  he  could  not  find  in  mere  reading.  The  great  modern 
work  in  the  theory  of  functions  and  in  allied  disciplines,  he 
never  became  acquainted  with  .  .  . 

What  would  have  been  the  effect  if,  in  the  prime  of  his  powers, 
he  had  been  surrounded  by  the  influences  which  prevail  in  Berlin 
or  in  Gottingen?  It  may  be  confidently  taken  for  granted  that 
he  would  have  done  splendid  work  in  those  domains  of  analysis, 
which  have  furnished  the  laurels  of  the  great  mathematicians  of 
Germany  and  France  in  the  second  half  of  the  present  century. 

FRANKLIN,  F 

Johns  Hopkins  University  Circulars  16  (1897), 
p.  54. 


PERSONS   AND   ANECDOTES  173 

1034.  If  we  survey  the  mathematical  works  of  Sylvester,  we 
recognize  indeed  a  considerable  abundance,  but  in  contradistinc- 
tion to  Cayley — not  a  versatility  toward  separate  fields,  but, 
with  few  exceptions — a  confinement  to   arithmetic-algebraic 
branches.  .  .  . 

The  concept  of  Function  of  a  continuous  variable,  the  funda- 
mental concept  of  modern  mathematics,  plays  no  role,  is  indeed 
scarcely  mentioned  in  the  entire  work  of  Sylvester — Sylvester 
was  combinatorist  [combinatoriker]. — NOETHEB,  M. 

Mathematische  Annalen,  Bd.50  (1 898),  pp.1 34~ 

135. 

1035.  Sylvester's  methods!  He  had  none.    "Three  lectures  will 
be  delivered  on  a  New  Universal  Algebra,"  he  would  say;  then, 
"  The  course  must  be  extended  to  twelve."    It  did  last  all  the 
rest  of  that  year.     The  following  year  the  course  was  to  be 
Substitutions-Theorie,  by  Netto.     We  all  got  the  text.     He 
lectured  about  three  times,  following  the  text  closely  and  stop- 
ping sharp  at  the  end  of  the  hour.   Then  he  began  to  think  about 
matrices  again.     "  I  must  give  one  lecture  a  week  on  those," 
he  said.    He  could  not  confine  himself  to  the  hour,  nor  to  the 
one  lecture  a  week.    Two  weeks  were  passed,  and  Netto  was 
forgotten  entirely  and  never  mentioned  again.    Statements  like 
the  following  were  not  unfrequent  in  his  lectures:  "  I  haven't 
proved  this,  but  I  am  as  sure  as  I  can  be  of  anything  that  it 
must  be  so.    From  this  it  will  follow,  etc."    At  the  next  lecture  it 
turned  out  that  what  he  was  so  sure  of  was  false.    Never  mind, 
he  kept  on  forever  guessing  and  trying,  and  presently  a  wonder- 
ful discovery  followed,  then  another  and  another.    Afterward  he 
would  go  back  and  work  it  all  over  again,  and  surprise  us  with 
all  sorts  of  side  lights.    He  then  made  another  leap  in  the  dark, 
more  treasures  were  discovered,  and  so  on  forever. 

DAVIS,  E.  W. 

Cajori's  Teaching  and  History  of  Mathematics 
in  the  U.  S.  (Washington,  1890),  pp.  265-266. 

1036.  I  can  see  him  [Sylvester]  now,  with  his  white  beard 
and  few  locks  of  gray  hair,  his  forehead  wrinkled  o'er  with 
thoughts,  writing  rapidly  his  figures  and  formulae  on  the  board, 
sometimes  explaining  as  he  wrote,  while  we,  his  listeners,  caught 


174  MEMORABILIA   MATHEMATICA 

the  reflected  sounds  from  the  board.  But  stop,  something  is  not 
right,  he  pauses,  his  hand  goes  to  his  forehead  to  help  his  thought, 
he  goes  over  the  work  again,  emphasizes  the  leading  points,  and 
finally  discovers  his  difficulty.  Perhaps  it  is  some  error  in  his 
figures,  perhaps  an  oversight  in  the  reasoning.  Sometimes, 
however,  the  difficulty  is  not  elucidated,  and  then  there  is  not 
much  to  the  rest  of  the  lecture.  But  at  the  next  lecture  we 
would  hear  of  some  new  discovery  that  was  the  outcome  of  that 
difficulty,  and  of  some  article  for  the  Journal,  which  he  had 
begun.  If  a  text-book  had  been  taken  up  at  the  beginning,  with 
the  intention  of  following  it,  that  text-book  was  most  likely 
doomed  to  oblivion  for  the  rest  of  the  term,  or  until  the  class  had 
been  made  listeners  to  every  new  thought  and  principle  that  had 
sprung  from  the  laboratory  of  his  mind,  in  consequence  of  that 
first  difficulty.  Other  difficulties  would  soon  appear,  so  that  no 
text-book  could  last  more  than  half  of  the  term.  In  this  way 
his  class  listened  to  almost  all  of  the  work  that  subsequently 
appeared  in  the  Journal.  It  seemed  to  be  the  quality  of  his 
mind  that  he  must  adhere  to  one  subject.  He  would  think 
about  it,  talk  about  it  to  his  class,  and  finally  write  about  it  for 
the  Journal.  The  merest  accident  might  start  him,  but  once 
started,  every  moment,  every  thought  was  given  to  it,  and,  as 
much  as  possible,  he  read  what  others  had  done  in  the  same 
direction;  but  this  last  seemed  to  be  his  real  point;  he  could  not 
read  without  finding  difficulties  in  the  way  of  understanding  the 
author.  Thus,  often  his  own  work  reproduced  what  had  been 
done  by  others,  and  he  did  not  find  it  out  until  too  late. 

A  notable  example  of  this  is  in  his  theory  of  cyclotomic  func- 
tions, which  he  had  reproduced  in  several  foreign  journals,  only 
to  find  that  he  had  been  greatly  anticipated  by  foreign  authors. 
It  was  manifest,  one  of  the  critics  said,  that  the  learned  professor 
had  not  read  Kummer's  elementary  results  in  the  theory  of  ideal 
primes.  Yet  Professor  Smith's  report  on  the  theory  of  numbers, 
which  contained  a  full  synopsis  of  Kummer's  theory,  was 
Professor  Sylvester's  constant  companion. 

This  weakness  of  Professor  Sylvester,  in  not  being  able  to 
read  what  others  had  done,  is  perhaps  a  concomitant  of  his 
peculiar  genius.  Other  minds  could  pass  over  little  difficulties 
and  not  be  troubled  by  them,  and  so  go  on  to  a  final  under- 


PERSONS  AND  ANECDOTES  175 

standing  of  the  results  of  the  author.  But  not  so  with  him. 
A  difficulty,  however  small,  worried  him,  and  he  was  sure  to 
have  difficulties  until  the  subject  had  been  worked  over  in  his 
own  way,  to  correspond  with  his  own  mode  of  thought.  To 
read  the  work  of  others,  meant  therefore  to  him  an  almost 
independent  development  of  it.  Like  the  man  whose  pleasure  in 
life  is  to  pioneer  the  way  for  society  into  the  forests,  his  rugged 
mind  could  derive  satisfaction  only  in  hewing  out  its  own  paths; 
and  only  when  his  efforts  brought  him  into  the  uncleared  fields  of 
mathematics  did  he  find  his  place  in  the  Universe. 

HATHAWAY,  A.  S. 

F.  Cajori's  Teaching  and  History  of  Mathe- 
matics in  the  U.  S.  (Washington,  1890),  pp. 
266-267. 

1037.  Professor   Cayley  has   since   informed   me   that   the 
theorem  about  whose  origin  I  was  in  doubt,  will  be  found  in 
Schlafli's  "  De  Eliminatione."    This  is  not  the  first  unconscious 
plagiarism  I  have  been  guilty  of  towards  this  eminent  man 
whose  friendship  I  am  proud  to  claim.    A  more  glaring  case  oc- 
curs in  a  note  by  me  in  the  "  Comptes  Rendus,"  on  the  twenty- 
seven  straight  lines  of  cubic  surfaces,  where  I  believe  I  have 
followed  (like  one  walking  in  his  sleep),  down  to  the  very 
nomenclature  and  notation,  the  substance  of  a  portion  of  a 
paper   inserted  by  Schlafli   hi  the   "  Mathematical  Journal," 
which  bears  my  name  as  one  of  the  editors  upon  the  face. 

SYLVESTER,  J.  J. 

Philosophical  Transactions  of  the  Royal 
Society  (1864),  p.  642. 

1038.  He  [Sylvester]  had  one  remarkable  peculiarity.     He 
seldom    remembered   theorems,    propositions,    etc.,    but    had 
always  to  deduce  them  when  he  wished  to  use  them.    In  this 
he  was  the  very  antithesis  of  Cayley,  who  was  thoroughly  con- 
versant with  everything  that  had  been  done  in  every  branch  of 
mathematics. 

I  remember  once  submitting  to  Sylvester  some  investigations 
that  I  had  been  engaged  on,  and  he  immediately  denied  my 
first  statement,  saying  that  such  a  proposition  had  never  been 
heard  of,  let  alone  proved.  To  his  astonishment,  I  showed  him  a 


176  MEMORABILIA   MATHEMATICA 

paper  of  his  own  in  which  he  had  proved  the  proposition;  in 
fact,  I  believe  the  object  of  his  paper  had  been  the  very  proof 
which  was  so  strange  to  him. — DURFEE,  W.  P. 

F.  Cajori's  Teaching  and  History  of  Mathe- 
matics in  the  U.  S.  (Washington,  1890),  p.  268. 

1039.  A  short,  broad  man  of  tremendous  vitality,  the  physical 
type  of  Hereward,  the  last  of  the  English,  and  his  brother-in- 
arms, Winter,  Sylvester's  capacious  head  was  ever  lost  in  the 
highest  cloud-lands  of  pure  mathematics.    Often  in  the  dead  of 
night  he  would  get  his  favorite  pupil,  that  he  might  communi- 
cate the  very  last  product  of  his  creative  thought.    Everything 
he  saw  suggested  to  him  somethihg  new  in  the  higher  algebra. 
This  transmutation  of  everything  into  new  mathematics  was  a 
revelation  to  those  who  knew  him  intimately.    They  began  to 
do  it  themselves.    His  ease  and  fertility  of  invention  proved  a 
constant  encouragement,   while  his  contempt  for  provincial 
stupidities,  such  as  the  American  hieroglyphics  for  TT  and  e, 
which  have  even  found  their  way  into  Webster's  Dictionary, 
made*  each  young  worker  apply  to  himself  the  strictest  tests. 

HALSTED,  G.  B. 

F.  Cajori's  Teaching  and  History  of  Mathe- 
matics in  the  U.  S.  (Washington,  1890),  p.  265. 

1040.  Sylvester's  writings  are  flowery  and  eloquent.    He  was 
able  to  make  the  dullest  subject  bright,  fresh  and  interesting. 
His  enthusiasm  is  evident  in  every  line.    He  would  get  quite 
close  up  to  his  subject,  so  that  everything  else  looked  small  in 
comparison,  and  for  the  time  would  think  and  make  others 
think  that  the  world  contained  no  finer  matter  for  contempla- 
tion.   His  handwriting  was  bad,  and  a  trouble  to  his  printers. 
His  papers  were  finished  with  difficulty.     No  sooner  was  the 
manuscript  in  the  editor's  hands  than  alterations,  corrections, 
ameliorations  and  generalizations  would  suggest  themselves  to 
his  mind,  and  every  post  would  carry  further  directions  to  the 
editors  and  printers. — MACMAHON.  P.  A. 

Nature,  Vol.  65  (1897),  p.  4^4- 

1041.  The  enthusiasm  of  Sylvester  for  his  own  work,  which 
manifests  itself  here  as  always,  indicates  one  of  his  characteristic 


PERSONS   AND   ANECDOTES  177 

qualities:  a  high  degree  of  subjectivity  in  his  productions  and 
publications.  Sylvester  was  so  fully  possessed  by  the  matter 
which  for  the  time  being  engaged  his  attention,  that  it  appeared 
to  him  and  was  designated  by  him  as  the  summit  of  all  that  is 
important,  remarkable  and  full  of  future  promise.  It  would 
excite  his  phantasy  and  power  of  imagination  in  even  a  greater 
measure  than  his  power  of  reflection,  so  much  so  that  he  could 
never  marshal  the  ability  to  master  his  subject-matter,  much 
less  to  present  it  in  an  orderly  manner. 

Considering  that  he  was  also  somewhat  of  a  poet,  it  will  be 
easier  to  overlook  the  poetic  flights  which  pervade  his  writing, 
often  bombastic,  sometimes  furnishing  apt  illustrations;  more 
damaging  is  the  complete  lack  of  form  and  orderliness  of  his 
publications  and  their  sketchlike  character,  .  .  .  which  must 
be  accredited  at  least  as  much  to  lack  of  objectivity  as  to  a 
superfluity  of  ideas.    Again,  the  text  is  permeated  with  asso- 
ciated emotional  expressions,  bizarre  utterances  and  paradoxes 
and  is  everywhere  accompanied  by  notes,  which  constitute  an 
essential  part  of  Sylvester's  method  of  presentation,  embodying 
relations,  whether  proximate  or  remote,  which  momentarily 
suggested  themselves.     These  notes,  full  of  inspiration  and 
occasional  flashes  of  genius,  are  the  more  stimulating  owing  to 
their  incompleteness.    But  none  of  his  works  manifest  a  desire  to 
penetrate  the  subject  from  all  sides  and  to  allow  it  to  mature; 
each  mere  surmise,  conceptions  which  arose  during  publication, 
immature  thoughts  and  even  errors  were  ushered  into  publicity 
at  the  moment  of  their  inception,  with  utmost  carelessness,  and 
always  with  complete  unfamiliarity  of  the  literature  of  the 
subject.    Nowhere  is  there  the  least  trace  of  self-criticism.    No 
one  can  be  expected  to  read  the  treatises  entire,  for  in  the  form 
in  which  they  are  available  they  fail  to  give  a  clear  view  of  the 
matter  under  contemplation. 

Sylvester's  was  not  a  harmoniously  gifted  or  well-balanced 
mind,  but  rather  an  instinctively  active  and  creative  mind,  free 
from  egotism.  His  reasoning  moved  hi  generalizations,  was 
frequently  influenced  by  analysis  and  at  times  was  guided  even 
by  mystical  numerical  relations.  His  reasoning  consists  less 
frequently  of  pure  intelligible  conclusions  than  of  inductions,  or 
rather  conjectures  incited  by  individual  observations  and 


178  MEMORABILIA   MATHEMATICA 

verifications.  In  this  he  was  guided  by  an  algebraic  sense, 
developed  through  long  occupation  with  processes  of  forms,  and 
this  led  him  luckily  to  general  fundamental  truths  which  in  some 
instances  remain  veiled.  His  lack  of  system  is  here  offset  by  the 
advantage  of  freedom  from  purely  mechanical  logical  activity. 

The  exponents  of  his  essential  characteristics  are  an  intuitive 
talent  and  a  faculty  of  invention  to  which  we  owe  a  series  of 
ideas  of  lasting  value  and  bearing  the  germs  of  fruitful  methods. 
To  no  one  more  fittingly  than  to  Sylvester  can  be  applied  one  of 
the  mottos  of  the  Philosophic  Magazine: 

"Admiratio  generat  quaestionem,  quaestio  investigationem 
investigatio  inventionem." — NOETHER,  M. 

Mathematische  Annalen,  Bd.  50  (1898),  pp. 
155-160. 

1042.  Perhaps  I  may  without  immodesty  lay  claim  to  the 
appellation  of  Mathematical  Adam,  as  I  believe  that  I  have 
given  more  names   (passed  into  general  circulation)   of  the 
creatures  of  the  mathematical  reason  than  all  the  other  mathe- 
maticians of  the  age  combined. — SYLVESTER,  J.  J. 

Nature,  Vol.  87  (1887-1888),  p.  162. 

1043.  Tait  dubbed  Maxwell  dp/dt,  for  according  to  ther- 
modynamics dp/dt  =  JCM  (where  C  denotes  Carnot's  func- 
tion) the  initials  of  (J.  C.)  Maxwell's  name.    On  the  other  hand 
Maxwell  denoted  Thomson  by  T  and  Tait  by  T';  so  that  it 
became  customary  to  quote  Thomson  and  Tait's  Treatise  on 
Natural  Philosophy  as  T  and  T'. — MACFARLANE,  A. 

Bibliotheca  Mathematica,  Bd.  3  (1903),  p.  187. 

1044.  In  future  times  Tait  will  be  best  known  for  his  work  in 
the  quaternion  analysis.    Had  it  not  been  for  his  expositions, 
developments  and  applications,  Hamilton's  invention  would  be 
today,  hi  all  probability,  a  mathematical  curiosity. 

MACFARLANE,  A. 
Bibliotheca  Mathematica,  Bd.  3  (1903),  p.  189. 

1045.  Not  seldom  did  he  [Sir  William  Thomson],   in  his 
writings,  set  down  some  mathematical  statement  with  the 
prefacing  remark  "  it  is  obvious  that "  to  the  perplexity  of  mathe- 


PERSONS  AND  ANECDOTES  179 

matical  readers,  to  whom  the  statement  was  anything  but 
obvious  from  such  mathematics  as  preceded  it  on  the  page. 
To  him  it  was  obvious  for  physical  reasons  that  might  not  sug- 
gest themselves  at  all  to  the  mathematician,  however  competent. 

THOMPSON,  S.  P. 
Life  of  Lord  Kelvin  (London,  1910),  p.  1186. 

1046.  The  following  is  one  of  the  many  stories  told  of  "old 
Donald   McFarlane"    the   faithful   assistant   of   Sir   William 
Thomson. 

The  father  of  a  new  student  when  bringing  him  to  the  Uni- 
versity, after  calling  to  see  the  Professor  [Thomson]  drew  his 
assistant  to  one  side  and  besought  him  to  tell  him  what  his  son 
must  do  that  he  might  stand  well  with  the  Professor.  "You 
want  your  son  to  stand  weel  with  the  Profeessorr? "  asked 
McFarlane.  "Yes."  "Weel,  then,  he  must  just  have  a  guid 
bellyful  o'  mathematics!" — THOMPSON,  S.  P. 

Life  of  Lord  Kelvin  (London,  1910),  p.  4%0. 

1047.  The  following  story  (here  a  little  softened  from  the 
vernacular)  was  narrated  by  Lord  Kelvin  himself  when  dining  at 
Trinity  Hall:— 

A  certain  rough  Highland  lad  at  the  university  had  done 
exceedingly  well,  and  at  the  close  of  the  session  gained  prizes 
both  in  mathematics  and  in  metaphysics.  His  old  father  came 
up  from  the  farm  to  see  his  son  receive  the  prizes,  and  visited  the 
College.  Thomson  was  deputed  to  show  him  round  the  place. 
"Weel,  Mr.  Thomson,"  asked  the  old  man,  "and  what  may  these 
mathematics  be,  for  which  my  son  has  getten  a  prize?  "  "I  told 
him,"  replied  Thomson,  "  that  mathematics  meant  reckon- 
ing with  figures,  and  calculating."  "Oo  ay,"  said  the  old  man, 
"he'll  ha'  getten  that  fra'  me:  I  were  ever  a  braw  hand  at  the 
countin'."  After  a  pause  he  resumed:  "And  what,  Mr.  Thom- 
son, might  these  metapheesics  be?"  "  I  endeavoured,"  replied 
Thomson,  "  to  explain  how  metaphysics  was  the  attempt  to 
express  in  language  the  indefinite."  The  old  Highlander  stood 
still  and  scratched  his  head.  "Oo  ay:  may  be  he'll  ha'  getten 
that  fra '  his  mither.  She  were  aye  a  bletherin '  body." 

THOMPSON,  S.  P. 
Life  of  Lord  Kelvin  (London,  1910),  p.  1124- 


180  MEMORABILIA   MATHEMATICA 

1048.  Lord  Kelvin,  unable  to  meet  his  classes  one  day,  posted 
the  following  notice  on  the  door  of  his  lecture  room, — 

"Professor  Thomson  will  not  meet  his  classes  today." 
The  disappointed  class  decided  to  play  a  joke  on  the  professor. 
Erasing  the  "  c  "  they  left  the  legend  to  read, — 

"Professor  Thomson  will  not  meet  his  lasses  today." 
When  the  class  assembled  the  next  day  in  anticipation  of  the 
effect  of  their  joke,  they  were  astonished  and  chagrined  to  find 
that  the  professor  had  outwitted  them.    The  legend  of  yesterday 
was  now  found  to  read, — 

"Professor  Thomson  will  not  meet  his  asses  today."  * 

NORTHKUP,  CYRUS. 

University  of  Washington  Address,  November  2, 
1908. 

1049.  One  morning  a  great  noise  proceeded  from  one  of  the 
classrooms  [of  the  Braunsberger  gymnasium]  and  on  investiga- 
tion it  was  found  that  Weierstrass,  who  was  to  give  the  recita- 
tion, had  not  appeared.    The  director  went  in  person  to  Weier- 
strass '  dwelling  and  on  knocking  was  told  to  come  in.   There  sat 
Weierstrass  by  a  glimmering  lamp  in  a  darkened  room  though 
it  was  daylight  outside.    He  had  worked  the  night  through  and 
had  not  noticed  the  approach  of  daylight.    When  the  director 
reminded  him  of  the  noisy  throng  of  students  who  were  waiting 
for  him,  his  only  reply  was  that  he  could  impossibly  interrupt 
his  work;  that  he  was  about  to  make  an  important  discovery 
which  would  attract  attention  in  scientific  circles. — LAMPE,  E. 

Karl  Weierstrass:  Jahrbuch  der  Deutschen 
Mathematiker  Vereinigung,  Bd.  6  (1897), 
pp.  88-89. 

1050.  Weierstrass  related  .  .  .  that  he  followed  Sylvester's 
papers  on  the  theory  of  algebraic  forms  very  attentively  until 
Sylvester  began  to  employ  Hebrew  characters.    That  was  more 

than  he  could  stand  and  after  that  he  quit  him. — LAMPE,  E. 
Naturwissenschaftliche    Rundschau,    Bd.    12 
(1897),  p.  361. 

*  Author's  note.  My  colleague,  Dr.  E.  T.  Bell,  informs  me  that  this 
same  anecdote  is  associated  with  the  name  of  J.  S.  Blackie,  Professor 
of  Greek  at  Aberdeen  and  Edinburgh. 


CHAPTER  XI 

MATHEMATICS  AS  A  FINE  ART 

1101.  The  world  of  idea  which  it  discloses  or  illuminates,  the 
contemplation  of  divine  beauty  and  order  which  it  induces,  the 
harmonious  connexion  of  its  parts,  the  infinite  hierarchy  and 
absolute  evidence  of  the  truths  with  which  it  is  concerned, 
these,  and  such  like,  are  the  surest  grounds  of  the  title  of  mathe- 
matics to  human  regard,  and  would  remain  unimpeached  and 
unimpaired  were  the  plan  of  the  universe  unrolled  like  a  map  at 
our  feet,  and  the  mind  of  man  qualified  to  take  in  the  whole 
scheme  of  creation  at  a  glance. — SYLVESTER,  J.  J. 

Presidential  Address,  British  Association  Re- 
port (1869);  Collected  Mathematical  Papers, 
Vol.  2,  -p.  659. 

1102.  Mathematics  has  a  triple  end.    It  should  furnish  an 
instrument  for  the  study  of  nature.     Furthermore  it  has  a 
philosophic  end,  and,  I  venture  to  say,  an  end  esthetic.     It 
ought  to  incite  the  philosopher  to  search  into  the  notions  of 
number,  space,  and  time;  and,  above  all,  adepts  find  in  mathe- 
matics delights  analogous  to  those  that  painting  and  music  give. 
They  admire  the  delicate  harmony  of  number  and  of  forms;  they 
are  amazed  when  a  new  discovery  discloses  for  them  an- un- 
locked for  perspective;  and  the  joy  they  thus  experience,  has 
it  not  the  esthetic  character  although  the  senses  take  no  part  in 
it?    Only  the  privileged  few  are  called  to  enjoy  it  fully,  it  is 
true;  but  is  it  not  the  same  with  all  the  noblest  arts?    Hence  I 
do  not  hesitate  to  say  that  mathematics  deserves  to  be  culti- 
vated for  its  own  sake,  and  that  the  theories  not  admitting  of 
application  to  physics  deserve  to  be  studied  as  well  as  others. 

POINCARE,  HENRI. 

The  Relation  of  Analysis  and  Mathematical 
Physics;    Bulletin    American    Mathematical 
Society,  Vol.  4  (1899),  p.  248. 
181 


182  MEMORABILIA   MATHEMATICA 

1103.  I  like  to  look  at  mathematics  almost  more  as  an  art  than 
as  a  science;  for  the  activity  of  the  mathematician,  constantly 
creating  as  he  is,  guided  though  not  controlled  by  the  external 
world  of  the  senses,  bears  a  resemblance,  not  fanciful  I  believe 
but  real,  to  the  activity  of  an  artist,  of  a  painter  let  us  say. 
Rigorous  deductive  reasoning  on  the  part  of  the  mathematician 
may  be  likened  here  to  technical  skill  hi  drawing  on  the  part 
of  the  painter.    Just  as  no  one  can  become  a  good  painter  with- 
out a  certain  amount  of  skill,  so  no  one  can  become  a  mathe- 
matician without  the  power  to  reason  accurately  up  to  a  certain 
point.    Yet  these  qualities,  fundamental  though  they  are,  do  not 
make  a  painter  or  mathematician  worthy  of  the  name,  nor  in- 
deed are  they  the  most  important  factors  in  the  case.    Other 
qualities  of  a  far  more  subtle  sort,  chief  among  which  in  both 
cases  is  imagination,  go  to  the  making  of  a  good  artist  or  good 
mathematician. — BOCHER,  MAXIME. 

Fundamental  Conceptions  and  Methods  in 
Mathematics;  Bulletin  American  Mathematical 
Society,  Vol.  9  (1904),  P- 133. 

1104.  Mathematics,  rightly  viewed,  possesses  not  only  truth, 
but  supreme  beauty — a  beauty  cold  and  austere,  like  that  of 
sculpture,  without  appeal  to  any  part  of  our  weaker  nature, 
without  the  gorgeous  trappings  of  painting  or  music,  yet  sub- 
limely pure,  and  capable  of  a  stern  perfection  such  as  only  the 
greatest  art  can  show.    The  true  spirit  of  delight,  the  exaltation, 
the  sense  of  being  more  than  man,  which  is  the  touchstone  of  the 
highest  excellence,  is  to  be  found  in  mathematics  as  surely  as  in 
poetry.    What  is  best  in  mathematics  deserves  not  merely  to  be 
learned  as  a  task,  but  to  be  assimilated  as  a  part  of  daily  thought, 
and  brought  again  and  again  before  the  mind  with  ever-renewed 
encouragement.    Real  life  is,  to  most  men,  a  long  second-best,  a 
perpetual  compromise  between  the  real  and  the  possible;  but 
the  world  of  pure  reason  knows  no  compromise,  no  practical 
limitations,  no  barrier  to  the  creative  activity  embodying  in 
splendid  edifices  the  passionate  aspiration  after  the  perfect 
from  which  all  great  work  springs.    Remote  from  human  pas- 
sions, remote  even  from  the  pitiful  facts  of  nature,  the  genera- 
tions have  gradually  created  an  ordered  cosmos,  where  pure 
thought  can  dwell  as  in  its  natural  home,  and  where  one,  at 


MATHEMATICS   AS   A    FINE   ART  183 

least,  of  our  nobler  impulses  can  escape  from  the  dreary  exile  of 
the  natural  world. — RUSSELL,  BERTRAND. 

The  Study  of  Mathematics:  Philosophical 
Essays  (London,  1910),  p.  78. 

1105.  It  was  not  alone  the  striving  for  universal  culture  which 
attracted  the  great  masters  of  the  Renaissance,  such  as  Brunel- 
lesco,  Leonardo  de  Vinci,  Raphael,  Michael  Angelo  and  espe- 
cially Albrecht  Diirer,  with  irresistible  power  to  the  mathe- 
matical  sciences.      They   were   conscious  that,  with  all  the 
freedom  of  the  individual  phantasy,  art  is  subject  to  necessary 
laws,  and  conversely,  with  all  its  rigor  of  logical  structure, 
mathematics  follows  esthetic  laws. — RUDIO,  F. 

Virchow-Holtzendorf:  Sammlung  gemeinver- 
stdndliche  wissenschaftliche  Vortrage,  Heft  1^2, 
p.  19. 

1106.  Surely  the  claim  of  mathematics  to  take  a  place  among 
the  liberal  arts  must  now  be  admitted  as  fully  made  good. 
Whether  we  look  at  the  advances  made  in  modern  geometry,  in 
modern  integral  calculus,  or  in  modern  algebra,  in  each  of  these 
three  a  free  handling  of  the  material  employed  is  now  possible, 
and  an  almost  unlimited  scope  is  left  to  the  regulated  play  of 
fancy.    It  seems  to  me  that  the  whole  of  aesthetic  (so  far  as  at 
present  revealed)  may  be  regarded  as  a  scheme  having  four 
centres,  which  may  be  treated  as  the  four  apices  of  a  tetrahe- 
dron, namely  Epic,  Music,  Plastic,  and  Mathematic.    There  will 
be  found  a  common  plane  to  every  three  of  these,  outside  of 
which  lies  the  fourth;  and  through  every  two  may  be  drawn  a 
common  axis  opposite  to  the  axis  passing  through  the  other  two. 
So  far  is  certain  and  demonstrable.    I  think  it  also  possible  that 
there  is  a  centre  of  gravity  to  each  set  of  three,  and  that  the 
line  joining  each  such  centre  with  the  outside  apex  will  intersect 
in  a  common  point — the  centre  of  gravity  of  the  whole  body  of 
aesthetic;  but  what  that  centre  is  or  must  be  I  have  not  had 
time  to  think  out. — SYLVESTER,  J.  J. 

Proof  of  the  hitherto  undemonstrated  Funda- 
mental Theorem  of  Invariants:  Collected  Mathe- 
matical Papers,  Vol.  8,  p.  128. 


184  MEMORABILIA   MATHEMATICA 

1107.  It  is  with  mathematics  not  otherwise  than  it  is  with 
music,  painting  or  poetry.   Anyone  can  become  a  lawyer,  doctor 
or  chemist,  and  as  such  may  succeed  well,  provided  he  is  clever 
and  industrious,  but  not  every  one  can  become  a  painter,  or  a 
musician,  or  a  mathematician:  general  cleverness  and  industry 
alone  count  here  for  nothing. — MOEBIUS,  P.  J. 

Ueber  die  Anlage  zur  Mathematik  (Leipzig, 
1900),  p.  5. 

1108.  The  true  mathematician  is  always  a  good  deal  of  an 
artist,  an  architect,  yes,  of  a  poet.    Beyond  the  real  world, 
though  perceptibly  connected  with  it,  mathematicians  have 
intellectually  created  an  ideal  world,  which  they  attempt  to 
develop  into  the  most  perfect  of  all  worlds,  and  which  is  being 
explored  in  every  direction.   None  has  the  faintest  conception  of 
this  world,  except  he  who  knows  it. — PEINGSHEIM,  A. 

Jahresbericht  der  Deutschen  Mathematiker 
Vereinigung,  Bd.  32,  p.  881. 

1109.  Who  has  studied  the  works  of  such  men  as  Euler, 
Lagrange,  Cauchy,  Riemann,  Sophus  Lie,  and  Weierstrass,  can 
doubt  that  a  great  mathematician  is  a  great  artist?    The  facul- 
ties possessed  by  such  men,  varying  greatly  in  kind  and  degree 
with  the  individual,  are  analogous  with  those  requisite  for  con- 
structive art.    Not  every  mathematician  possesses  in  a  specially 
high  degree  that  critical  faculty  which  finds  its  employment  in 
the  perfection  of  form,  hi  conformity  with  the  ideal  of  logical 
completeness;  but  every  great  mathematician  possesses  the  rarer 
faculty  of  constructive  imagination. — HOBSON,  E.  W. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science  (1910)  Nature, 
Vol.  84,  p.  290. 

1110.  Mathematics  has  beauties  of  its  own — a  symmetry  and 
proportion  hi  its  results,  a  lack  of  superfluity,  an  exact  adapta- 
tion of  means  to  ends,  which  is  exceedingly  remarkable  and  to 
be  found  elsewhere  only  hi  the  works  of  the  greatest  beauty. 
It  was  a  felicitous  expression  of  Goethe's  to  call  a  noble  cathe- 
dral "frozen  music,"  but  it  might  even  better  be  called  "petri- 
fied mathematics."     The  beauties  of  mathematics — of  sim- 


MATHEMATICS   AS   A    FINE   ART  185 

plicity,  of  symmetry,  of  completeness — can  and  should  be 
exemplified  even  to  young  children.  When  this  subject  is 
properly  and  concretely  presented,  the  mental  emotion  should 
be  that  of  enjoyment  of  beauty,  not  that  of  repulsion  from  the 
ugly  and  the  unpleasant. — YOUNG,  J.  W.  A. 

The   Teaching  of  Mathematics   (New   York, 

1907),  p.  44- 

1111.  A  peculiar  beauty  reigns  in  the  realm  of  mathematics,  a 
beauty  which  resembles  not  so  much  the  beauty  of  art  as  the 
beauty  of  nature  and  which  affects  the  reflective  mind,  which 
has  acquired  an  appreciation  of  it,  very  much  like  the  latter. 

KUMMER,  E.  E. 
Berliner  Monatsberichte  (1867),  p.  395. 

1112.  Mathematics  make  the  mind  attentive  to  the  objects 
which  it  considers.     This  they  do  by  entertaining  it  with  a 
great  variety  of  truths,  which  are  delightful  and  evident,  but  not 
obvious.    Truth  is  the  same  thing  to  the  understanding  as  music 
to  the  ear  and  beauty  to  the  eye.    The  pursuit  of  it  does  really  as 
much  gratify  a  natural  faculty  implanted  in  us  by  our  wise 
Creator  as  the  pleasing  of  our  senses :  only  in  the  former  case,  as 
the  object  and  faculty  are  more  spiritual,the  delight  is  more  pure, 
free  from  regret,  turpitude,  lassitude,  and  intemperance  that 
commonly  attend  sensual  pleasures. — ARBUTHNOT,  JOHN. 

Usefulness  of  Mathematical  Learning. 

1113.  However  far  the  calculating  reason  of  the  mathe- 
matician may  seem  separated  from  the  bold  flight  of  the  artist's 
phantasy,  it  must  be  remembered  that  these  expressions  are  but 
momentary  images  snatched  arbitrarily  from  among  the  ac- 
tivities of  both.    In  the  projection  of  new  theories  the  mathe- 
matician needs  as  bold  and  creative  a  phantasy  as  the  productive 
artist,  and  in  the  execution  of  the  details  of  a  composition  the 
artist  too  must  calculate  dispassionately  the  means  which  are 
necessary  for  the  successful  consummation  of  the  parts.    Com- 
mon to  both  is  the  creation,  the  generation,  of  forms  out  of 
mind. — LAMPE,  E. 

Die  Entwickelung  der  Mathematik,  etc.  (Berlin, 
1898),  p.  4- 


186  MEMORABILIA    MATHEMATICA 

1114.  As  pure  truth  is  the  polar  star  of  our  science  [mathe- 
matics], so  it  is  the  great  advantage  of  our  science  over  others 
that  it  awakens  more  easily  the  love  of  truth  in  our  pupils.  .  .  . 
If  Hegel  justly  said,  "  Whoever  does  not  know  the  works  of  the 
ancients,  has  lived  without  knowing  beauty,"  Schellbach  re- 
sponds with  equal  right,  "  Who  does  not  know  mathematics, 
and  the  results  of  recent  scientific  investigation,  dies  without 

knowing  truth."  —  SIMON,  MAX. 

Quoted  in  J.W.  A.  Young:  Teaching  of  Mathe- 
matics (New  York,  1907),  p.  44- 

1115.  Biichsel  in  his  reminiscences  from  the  life  of  a  country 
parson  relates  that  he  sought  his  recreation  in  Lacroix's  Dif- 
ferential Calculus  and  thus  found  intellectual  refreshment  for 
his  calling.     Instances  like  this  make  manifest  the  great  ad- 
vantage which  occupation  with  mathematics  affords  to  one  who 
lives  remote  from  the  city  and  is  compelled  to  forego  the  pleas- 
ures of  art.     The  entrancing  charm  of  mathematics,  which 
captivates  every  one  who  devotes  himself  to  it,  and  which  is 
comparable  to  the  fine  frenzy  under  whose  ban  the  poet  com- 
pletes his  work,  has  ever  been  incomprehensible  to  the  spectator 
and  has  often  caused  the  enthusiastic  mathematician  to  be 
held  in  derision.    A  classic  illustration  is  the  example  of  Archi- 
medes, .  .  .  —  LAMPE,  E. 

Die  Entunckelung  der  Mathematik,  etc.  (Berlin 


1116.  Among  the  memoirs  of  Kirchhoff  are  some  of  uncom- 
mon beauty.  Beauty,  I  hear  you  ask,  do  not  the  Graces  flee 
where  integrals  stretch  forth  their  necks?  Can  anything  be 
beautiful,  where  the  author  has  no  time  for  the  slightest  external 
embellishment?  .  .  .  Yet  it  is  this  very  simplicity,  the  indis- 
pensableness  of  each  word,  each  letter,  each  little  dash,  that 
among  all  artists  raises  the  mathematician  nearest  to  the 
World-creator;  it  establishes  a  sublimity  which  is  equalled  in  no 
other  art,  —  something  like  it  exists  at  most  in  symphonic  music. 
The  Pythagoreans  recognized  already  the  similarity  between  the 
most  subjective  and  the  most  objective  of  the  arts.  .  .  .  Ultima 
se  tangunt.  How  expressive,  how  nicely  characterizing  withal  is 
mathematics!  As  the  musician  recognizes  Mozart,  Beethoven, 
Schubert  in  the  first  chords,  so  the  mathematician  would  dis- 


MATHEMATICS   AS   A   FINE   ART  187 

tinguish  his  Cauchy,  Gauss,  Jacobi,  Helmholtz  in  a  few  pages. 
Extreme  external  elegance,  sometimes  a  somewhat  weak  skeleton 
of  conclusions  characterizes  the  French;  the  English,  above  all 
Maxwell,  are  distinguished  by  the  greatest  dramatic  bulk. 
Who  does  not  know  Maxwell's  dynamic  theory  of  gases?  At 
first  there  is  the  majestic  development  of  the  variations  of 
velocities,  then  enter  from  one  side  the  equations  of  condition 
and  from  the  other  the  equations  of  central  motions, — higher 
and  higher  surges  the  chaos  of  formulas, — suddenly  four  words 
burst  forth:  "Put  n  =  5."  The  evil  demon  V  disappears  like 
the  sudden  ceasing  of  the  basso  parts  in  music,  which  hitherto 
wildly  permeated  the  piece;  what  before  seemed  beyond  con- 
trol is  now  ordered  as  by  magic.  There  is  no  time  to  state 
why  this  or  that  substitution  was  made,  he  who  cannot  feel  the 
reason  may  as  well  lay  the  book  aside;  Maxwell  is  no  program- 
musician  who  explains  the  notes  of  his  composition.  Forthwith 
the  formulas  yield  obediently  result  after  result,  until  the 
temperature-equilibrium  of  a  heavy  gas  is  reached  as  a  sur- 
prising final  climax  and  the  curtain  drops.  .  .  . 

Kirchhoff's  whole  tendency,  and  its  true  counterpart,  the 
form  of  his  presentation,  was  different.  .  .  .  He  is  charac- 
terized by  the  extreme  precision  of  his  hypotheses,  minute 
execution,  a  quiet  rather  than  epic  development  with  utmost 
rigor,  never  concealing  a  difficulty,  always  dispelling  the  faintest 
obscurity.  To  return  once  more  to  my  allegory,  he  resembled 
Beethoven,  the  thinker  in  tones. — He  who  doubts  that  mathe- 
matical compositions  can  be  beautiful,  let  him  read  his  memoir 
on  Absorption  and  Emission  (Gesammelte  Abhandlungen,  Leip- 
zig, 1882,  p.  571-598)  or  the  chapter  of  his  mechanics  devoted  to 

Hydrodynamics. — BOLTZMANN,  L. 

Gustav  Robert  Kirchhoff  (Leipzig  1888),  pp.  28-30. 

1117.      On  poetry  and  geometric  truth, 

And  their  high  privilege  of  lasting  life, 
From  all  internal  injury  exempt, 
I  mused;  upon  these  chiefly:  and  at  length, 
My  senses  yielding  to  the  sultry  air, 
Sleep  seized  me,  and  I  passed  into  a  dream. 

WORDSWORTH. 
The  Prelude,  Bk.  5. 


188  MEMORABILIA   MATHEMATICA 

1118.  Geometry  seems  to  stand  for  all  that  is  practical, 
poetry  for  all  that  is  visionary,  but  in  the  kingdom  of  the  imagi- 
nation you  will  find  them  close  akin,  and  they  should  go  together 
as  a  precious  heritage  to  every  youth. — MILNEB,  FLORENCE. 

School  Review,  1898,  p.  114. 

1119.  The  beautiful  has  its  place  hi  mathematics  as  elsewhere. 
The  prose  of  ordinary  intercourse  and  of  business  correspond- 
ence might  be  held  to  be  the  most  practical  use  to  which  lan- 
guage is  put,  but  we  should  be  poor  indeed  without  the  literature 
of  imagination.     Mathematics  too  has  its  triumphs  of  the 
creative  imagination,  its  beautiful  theorems,  its  proofs  and 
processes  whose  perfection  of  form  has  made  them  classic. 
He  must  be  a  "practical"  man  who  can  see  no  poetry  hi  mathe- 
matics.— WHITE,  W.  F. 

A    Scrap-book    of   Elementary    Mathematics 
(Chicago,  1908),  p.  208. 

1120.  I  venture  to  assert  that  the  feelings  one  has  when  the 
beautiful  symbolism  of  the  infinitesimal  calculus  first  gets  a 
meaning,  or  when  the  delicate  analysis  of  Fourier  has  been 
mastered,  or  while  one  follows  Clerk  Maxwell  or  Thomson  into 
the  strange  world  of  electricity,  now  growing  so  rapidly  in  form 
and  being,  or  can  almost  feel  with  Stokes  the  pulsations  of  light 
that  gives  nature  to  our  eyes,  or  track  with  Clausius  the  courses 
of  molecules  we  can  measure,  even  if  we  know  with  certainty 
that  we  can  never  see  them — I  venture  to  assert  that  these  feel- 
ings are  altogether  comparable  to  those  aroused  in  us  by  an 
exquisite  poem  or  a  lofty  thought. — WORKMAN,  W.  P. 

F.  Spencer:  Aim  and  Practice  of  Teaching 
(New  York,  1897),  p.  194. 

1121.  It  is  an  open  secret  to  the  few  who  know  it,  but  a 
mystery  and  stumbling  block  to  the  many,  that  Science  and 
Poetry  are  own  sisters;  insomuch  that  in  those  branches  of 
scientific  inquiry  which  are  most  abstract,  most  formal,  and  most 
remote  from  the  grasp  of  the  ordinary  sensible  imagination,  a 
higher  power  of  imagination  akin  to  the  creative  insight  of  the 
poet  is  most  needed  and  most  fruitful  of  lasting  work. 

POLLOCK,  F. 

Clifford's  Lectures  and  Essays   (New   York, 
1901),  Vol.  1,  Introduction,  p.  1. 


MATHEMATICS   AS   A    FINE   AKT  189 

1122.  It  is  as  great  a  mistake  to  maintain  that  a  high  develop- 
ment of  the  imagination  is  not  essential  to  progress  in  mathe- 
matical studies  as  to  hold  with  Ruskin  and  others  that  science 
and  poetry  are  antagonistic  pursuits. — HOFFMAN,  F.  S. 

Sphere  of  Science  (London,  1898},  p.  107. 

1123.  We  have  heard  much  about  the  poetry  of  mathe- 
matics, but  very  little  of  it  has  as  yet  been  sung.    The  ancients 
had  a  juster  notion  of  their  poetic  value  than  we.    The  most 
distinct  and  beautiful  statements  of  any  truth  must  take  at  last 
the  mathematical  form.     We  might  so  simplify  the  rules  of 
moral  philosophy,  as  well  as  of  arithmetic,  that  one  formula 
would  express  them  both. — THOREAU,  H.  D. 

A  Week  on  the  Concord  and  Merrimac  Rivers 
(Boston,  1893),  p.  477. 

1124.  We  do  not  listen  with  the  best  regard  to  the  verses  of  a 
man  who  is  only  a  poet,  nor  to  his  problems  if  he  is  only  an 
algebraist;  but  if  a  man  is  at  once  acquainted  with  the  geometric 
foundation  of  things  and  with  their  festal  splendor,  his  poetry  is 
exact  and  his  arithmetic  musical. — EMERSON,  R.  W. 

Society  and  Solitude,  Chap.  7,   Works  and 
Days. 

1125.  Mathesis  and  Poetry  are  ...  the  utterance  of  the 
same  power  of  imagination,  only  that  in  the  one  case  it  is  ad- 
dressed to  the  head,  and  in  the  other,  to  the  heart. 

HILL,  THOMAS. 

North  American  Review,  Vol.  85,  p.  230. 

1126.  The  Mathematics  are  usually  considered  as  being  the 
very  antipodes  of  Poesy.    Yet  Mathesis  and  Poesy  are  of  the 
closest  kindred,  for  they  are  both  works  of  the  imagination. 
Poesy  is  a  creation,  a  making,  a  fiction;  and  the  Mathematics 
have  been  called,  by  an  admirer  of  them,  the  sublimest  and  most 
stupendous  of  fictions.    It  is  true,  they  are  not  only  paffrjcris } 
learning,  but  iroirja-K,  a  creation. — HILL,  THOMAS. 

North  American  Review,  Vol.  85,  p.  229. 


190  MEMORABILIA    MATHEMATICA 

1127.  Music  and  poesy  used  to  quicken  you : 
The  mathematics,  and  the  metaphysics, 

Fall  to  them  as  you  find  your  stomach  serves  you. 
No  profit  grows,  where  is  no  pleasure  ta  'en : — 
In  brief,  sir,  study  what  you  most  affect. 

SHAKESPEARE. 
Taming  of  the  Shrew,  Act  1 ,  Scene  1 . 

1128.  Music  has  much  resemblance  to  algebra. — NOVALIS. 

Schriften,  Teil2  (Berlin,  1901),  p.  549. 

1129.  I  do  present  you  with  a  man  of  mine, 
Cunning  in  music  and  in  mathematics, 
To  instruct  her  fully  in  those  sciences, 

Whereof,  I  know,  she  is  not  ignorant. — SHAKESPEARE. 

Taming  of  the  Shrew,  Act  2,  Scene  1 . 

1130.  Saturated  with  that  speculative  spirit  then  pervading 
the  Greek  mind,  he  [Pythagoras]  endeavoured  to  discover  some 
principle  of  homogeneity  in  the  universe.     Before  him,  the 
philosophers  of  the  Ionic  school  had  sought  it  in  the  matter  of 
things;  Pythagoras  looked  for  it  in  the  structure  of  things.    He 
observed  the  various  numerical  relations  or  analogies  between 
numbers  and  the  phenomena  of  the  universe.    Being  convinced 
that  it  was  in  numbers  and  their  relations  that  he  was  to  find 
the  foundation  to  true  philosophy,  he  proceeded  to  trace  the 
origin  of  all  things  to  numbers.    Thus  he  observed  that  musical 
strings  of  equal  lengths  stretched  by  weights  having  the  pro- 
portion of  ^,  |,  |,  produced  intervals  which  were  an  octave,  a 
fifth  and  a  fourth.     Harmony,  therefore,  depends  on  musical 
proportion;  it  is  nothing  but  a  mysterious  numerical  relation. 
Where  harmony  is,  there  are  numbers.    Hence  the  order  and 
beauty  of  the  universe  have  their  origin  in  numbers.    There 
are  seven  intervals  in  the  musical  scale,  and  also  seven  planets 
crossing  the  heavens.     The  same  numerical  relations  which 
underlie  the  former  must  underlie  the  latter.    But  where  num- 
ber is,  there  is  harmony.    Hence  his  spiritual  ear  discerned  in  the 
planetary  motions  a  wonderful  "Harmony  of  spheres." 

CAJORI,  F. 
History  of  Mathematics  (New  York,  1897),  p.  67. 


MATHEMATICS   AS   A    FINE   ART  191 

1131.  May  not  Music  be  described  as  the  Mathematic  of 
sense,  Mathematic  as  Music  of  the  reason?  the  soul  of  each  the 
same !    Thus  the  musician  feels  Mathematic,  the  mathematician 
thinks  Music, — Music  the  dream,   Mathematic  the  working 
life — each  to  receive  its  consummation  from  the  other  when  the 
human  intelligence,  elevated  to  its  perfect  type,  shall  shine 
forth  glorified  in  some  future  Mozart-Dirichlet  or  Beethoven- 
Gauss — a  union  already  not  indistinctly  foreshadowed  in  the 
genius  and  labours  of  a  Helmholtz! — SYLVESTER,  J.  J. 

On  Newton's  Rule  for  the  Discovery  of  Imag- 
inary Roots;  Collected  Mathematical  Papers, 
Vol.  2,  p.  419. 

1132.  Just  as  the  musician  is  able  to  form  an  acoustic  image 
of  a  composition  which  he  has  never  heard  played  by  merely 
looking  at  its  score,  so  the  equation  of  a  curve,  which  he  has 
never   seen,    furnishes   the   mathematician   with   a   complete 
picture  of  its  course.    Yea,  even  more:  as  the  score  frequently 
reveals  to  the  musician  niceties  which  would  escape  his  ear 
because  of  the  complication  and  rapid  change  of  the  auditory 
impressions,  so  the  insight  which  the  mathematician  gains  from 
the  equation  of  a  curve  is  much  deeper  than  that  which  is 
brought  about  by  a  mere  inspection  of  the  curve. 

PRINGSHEIM,  A. 

Jahresbericht  der  Deutschen  Mathematiker 
Vereiningung,  Bd.  13,  p.  864- 

1133.  Mathematics  and  music,  the  most  sharply  contrasted 
fields  of  scientific  activity  which  can  be  found,  and  yet  related, 
supporting  each  other,  as  if  to  show  forth  the  secret  connection 
which  ties  together  all  the  activities  of  our  mind,  and  which 
leads  us  to  surmise  that  the  manifestations  of  the  artist's  genius 
are  but  the  unconscious  expressions  of  a  mysteriously  acting 
rationality. — HELMHOLTZ,  H. 

Vortrage  und  Reden,  Bd.  1  (Braunschweig, 
1884),  P-  82. 

1134.  Among  all  highly  civilized  peoples  the  golden  age  of 
art  has  always  been  closely  coincident  with  the  golden  age  of  the 
pure  sciences,  particularly  with  mathematics,  the  most  ancient 
among  them. 


192  MEMORABILIA   MATHEMATICA 

This  coincidence  must  not  be  looked  upon  as  accidental,  but 
as  natural,  due  to  an  inner  necessity.  Just  as  art  can  thrive  only 
when  the  artist,  relieved  of  the  anxieties  of  existence,  can  listen 
to  the  inspirations  of  his  spirit  and  follow  in  their  lead,  so 
mathematics,  the  most  ideal  of  the  sciences,  will  yield  its 
choicest  blossoms  only  when  life's  dismal  phantom  dissolves  and 
fades  away,  when  the  striving  after  naked  truth  alone  predomi- 
nates, conditions  which  prevail  only  in  nations  while  in  the  prime 
of  their  development. — LAMPE,  E. 

Die  Entwickelung  der  Mathematik  etc.  (Berlin, 

1893),  p.  4- 

1136.  Till  the  fifteenth  century  little  progress  appears  to  have 
been  made  in  the  science  or  practice  of  music;  but  since  that  era 
it  has  advanced  with  marvelous  rapidity,  its  progress  being 
curiously  parallel  with  that  of  mathematics,  inasmuch  as  great 
musical  geniuses  appeared  suddenly  among  different  nations, 
equal  in  their  possession  of  this  special  faculty  to  any  that  have 
since  arisen.  As  with  the  mathematical  so  with  the  musical 
faculty — it  is  impossible  to  trace  any  connection  between  its 
possession  and  survival  in  the  struggle  for  existence. 

WALLACE,  A.  R. 
Darwinism,  Chap.  15. 

1136.  In  my  opinion,  there  is  absolutely  no  trustworthy 
proof  that  talents  have  been  improved  by  their  exercise  through 
the  course  of  a  long  series  of  generations.  The  Bach  family 
shows  that  musical  talent,  and  the  Bernoulli  family  that  mathe- 
matical power,  can  be  transmitted  from  generation  to  genera- 
tion, but  this  teaches  us  nothing  as  to  the  origin  of  such  talents. 
In  both  families  the  high-watermark  of  talent  lies,  not  at  the 
end  of  the  series  of  generations,  as  it  should  do  if  the  results  of 
practice  are  transmitted,  but  in  the  middle.  Again,  talents 
frequently  appear  in  some  member  of  a  family  which  has  not 
been  previously  distinguished. 

Gauss  was  not  the  son  of  a  mathematician;  Handel's  father 
was  a  surgeon,  of  whose  musical  powers  nothing  is  known; 
Titian  was  the  son  and  also  the  nephew  of  a  lawyer,  while  he 
and  his  brother,  Francesco  Vecellio,  were  the  first  painters  in  a 


MATHEMATICS   AS   A    FINE   ART  193 

family  which  produced  a  succession  of  seven  other  artists  with 
diminishing  talents.  These  facts  do  not,  however,  prove  that 
the  condition  of  the  nerve-tracts  and  centres  of  the  brain,  which 
determine  the  specific  talent,  appeared  for  the  first  time  in  these 
men:  the  appropriate  condition  surely  existed  previously  in  their 
parents,  although  it  did  not  achieve  expression.  They  prove,  as 
it  seems  to  me,  that  a  high  degree  of  endowment  in  a  special 
direction,  which  we  call  talent,  cannot  have  arisen  from  the 
experience  of  previous  generations,  that  is,  by  the  exercise  of 
the  brain  in  the  same  specific  direction. 

WEISMANN,  AUGUST. 

Essays  upon  Heredity  [A.  E.  Shipley],  (Ox- 
ford, 1891),  Vol.  1,  p.  97. 


CHAPTER  XII 

MATHEMATICS   AS  A   LANGUAGE 

1201.  The  new  mathematics  is  a  sort  of  supplement  to  lan- 
guage, affording  a  means  of  thought  about  form  and  quantity 
and  a  means  of  expression,  more  exact,  compact,  and  ready  than 
ordinary  language.    The  great  body  of  physical  science,  a  great 
deal  of  the  essential  facts  of  financial  science,  and  endless  social 
and  political  problems  are  only  accessible  and  only  thinkable  to 
those  who  have  had  a  sound  training  in  mathematical  analysis, 
and  the  time  may  not  be  very  remote  when  it  will  be  understood 
that  for  complete  initiation  as  an  efficient  citizen  of  one  of  the 
new  great  complex  world  wide  states  that  are  now  developing, 
it  is  as  necessary  to  be  able  to  compute,  to  think  in  averages  and 
maxima  and  minima,  as  it  is  now  to  be  able  to  read  and  to  write. 

WELLS,  H.  G. 

Mankind   in   the   Making    (London,    1904), 
pp. 191-192. 

1202.  Mathematical  language  is  not  only  the  simplest  and 
most  easily  understood  of  any,  but  the  shortest  also. 

BROUGHAM,  H.  L. 

Works   (Edinburgh,   1872},   Vol.  7,  p.  817. 

1203.  Mathematics  is  the  science  of  definiteness,  the  necessary 
vocabulary  of  those  who  know. — WHITE,  W.  F. 

A    Scrap-book    of   Elementary    Mathematics 
(Chicago,  1908),  p.  7. 

1204.  Mathematics,  too,  is  a  language,  and  as  concerns  its 
structure  and  content  it  is  the  most  perfect  language  which 
exists,  superior  to  any  vernacular;  indeed,  since  it  is  understood 
by  every  people,  mathematics  may  be  called  the  language  of 
languages.  Through  it,  as  it  were,  nature  herself  speaks;  through 
it  the  Creator  of  the  world  has  spoken,  and  through  it  the 
Preserver  of  the  world  continues  to  speak. — DILLMANN,  C. 

Die    Mathematik    die    Fackeltragerin    einer 
neutn  Zeit  (Stuttgart,  1889),  p.  5. 
194 


MATHEMATICS   AS   A   LANGUAGE  195 

1205.  Would  it  sound  too  presumptuous  to  speak  of  perception 
as  a  quintessence  of  sensation,  language  (that  is,  communicable 
thought)  of  perception,  mathematics  of  language?    We  should 
then  have  four  terms  differentiating  from  inorganic  matter  and 
from  each  other  the  Vegetable,  Animal,  Rational,  and  Super- 
sensual  modes  of  existence. — SYLVESTER,  J.  J. 

Presidential    Address,     British    Association; 
Collected  Mathematical  Papers,  Vol.  2,  p.  652. 

1206.  Little  could  Plato  have  imagined,  when,  indulging  his 
instinctive  love  of  the  true  and  beautiful  for  their  own  sakes,  he 
entered  upon  these  refined  speculations    and   revelled   in    a 
world  of  his  own  creation,  that  he  was  writing  the  grammar  of 
the  language  in  which  it  would  be  demonstrated  in  after  ages 
that  the  pages  of  the  universe  are  written. — SYLVESTER,  J.  J. 

A    Probationary   Lecture   on   Geometry;   Col- 
lected Mathematical  Papers,  Vol.  2,  p.  7. 

1207.  It  is  the  symbolic  language  of  mathematics  only  which 
has  yet  proved  sufficiently  accurate  and  comprehensive  to 
demand  familiarity  with  this  conception  of  an  inverse  process. 

VENN,  JOHN. 

Symbolic  Logic  (London  and  New  York,  1894), 
p.  74. 

1208.  Without  this  language  [mathematics]  most  of  the  inti- 
mate analogies  of  things  would  have  remained  forever  unknown 
to  us;  and  we  should  forever  have  been  ignorant  of  the  internal 
harmony  of  the  world,  which  is  the  only  true  objective  real- 
ity. .  .  . 

This  harmony  ...  is  the  sole  objective  reality,  the  only 
truth  we  can  attain;  and  when  I  add  that  the  universal  harmony 
of  the  world  is  the  source  of  all  beauty,  it  will  be  understood  what 
price  we  should  attach  to  the  slow  and  difficult  progress  which 
little  by  little  enables  us  to  know  it  better. — POINCARE,  H. 

The  Value  of  Science  [Hoisted]  Popular  Science 
Monthly,  1906,  pp.  195-196. 

1209.  The  most  striking  characteristic  of  the  written  lan- 
guage of  algebra  and  of  the  higher  forms  of  the  calculus  is  the 


196  MEMORABILIA   MATHEMATICA 

sharpness  of  definition,  by  which  we  are  enabled  to  reason  upon 
the  symbols  by  the  mere  laws  of  verbal  logic,  discharging  our 
minds  entirely  of  the  meaning  of  the  symbols,  until  we  have 
reached  a  stage  of  the  process  where  we  desire  to  interpret  our 
results.  The  ability  to  attend  to  the  symbols,  and  to  perform 
the  verbal,  visible  changes  in  the  position  of  them  permitted  by 
the  logical  rules  of  the  science,  without  allowing  the  mind  to  be 
perplexed  with  the  meaning  of  the  symbols  until  the  result  is 
reached  which  you  wish  to  interpret,  is  a  fundamental  part  of 
what  is  called  analytical  power.  Many  students  find  them- 
selves perplexed  by  a  perpetual  attempt  to  interpret  not  only 
the  result,  but  each  step  of  the  process.  They  thus  lose  much  of 
the  benefit  of  the  labor-saving  machinery  of  the  calculus  and  are, 
indeed,  frequently  incapacitated  for  using  it. — HILL,  THOMAS. 

Uses  of  Mathesis;  Bibliotheca  Sacra,  Vol.  82, 
p.  505. 

1210.  The  prominent  reason  why  a  mathematician  can  be 
judged  by  none  but  mathematicians,  is  that  he  uses  a  peculiar 
language.  The  language  of  mathesis  is  special  and  untranslat- 
able. In  its  simplest  forms  it  can  be  translated,  as,  for  instance, 
we  say  a  right  angle  to  mean  a  square  corner.  But  you  go  a 
little  higher  hi  the  science  of  mathematics,  and  it  is  impossible 
to  dispense  with  a  peculiar  language.  It  would  defy  all  the 
power  of  Mercury  himself  to  explain  to  a  person  ignorant  of  the 
science  what  is  meant  by  the  single  phrase  "functional  expo- 
nent." How  much  more  impossible,  if  we  may  say  so,  would  it 
be  to  explain  a  whole  treatise  like  Hamilton's  Quaternions,  in 
such  a  wise  as  to  make  it  possible  to  judge  of  its  value!  But  to 
one  who  has  learned  this  language,  it  is  the  most  precise  and 
clear  of  all  modes  of  expression.  It  discloses  the  thought  exactly 
as  conceived  by  the  writer,  with  more  or  less  beauty  of  form,  but 
never  with  obscurity.  It  may  be  prolix,  as  it  often  is  among 
French  writers;  may  delight  in  mere  verbal  metamorphoses,  as 
in  the  Cambridge  University  of  England;  or  adopt  the  briefest 
and  clearest  forms,  as  under  the  pens  of  the  geometers  of  our 
Cambridge;  but  it  always  reveals  to  us  precisely  the  writer's 
thought. — HILL,  THOMAS. 

North  American  Review,  Vol.  85,  pp.  22^-225. 


MATHEMATICS   AS   A   LANGUAGE  197 

1211.  The  domain,  over  which  the  language  of  analysis  ex- 
tends its  sway,  is,  indeed,  relatively  limited,  but  within  this 
domain  it  so  infinitely  excels  ordinary  language  that  its  at- 
tempt to  follow  the  former  must  be  given  up  after  a  few  steps. 
The  mathematician,  who  knows  how  to  think  in  this  mar- 
velously  condensed  language,  is  as  different  from  the  mechanical 
computer  as  heaven  from  earth. — PRINGSHEIM,  A. 

Jahresberichte    der    Deutschen    Mathematiker 
Vereinigung,  Bd.  13,  p.  867. 

1212.  The  results  of  systematic  symbolical  reasoning  must 
always  express  general  truths,  by  their  nature;  and  do  not,  for 
their  justification,  require  each  of  the  steps  of  the  process  to 
represent  some  definite  operation  upon  quantity.    The  absolute 
universality  of  the  interpretation  of  symbols  is  the  fundamental 
principle  of  their  use. — WHEWELL,  WILLIAM. 

The   Philosophy    of   the    Inductive   Sciences, 
Part  I,  Bk.  2,  chap.  12,  sect.  2  (London,  1858). 

1213.  Anyone  who  understands  algebraic  notation,  reads  at  a 
glance  in  an  equation  results  reached  arithmetically  only  with 
great  labour  and  pains. — COURNOT,  A. 

Theory  of  Wealth  [N.  T.  Bacon],  (New  York, 
1897),  p.  4- 

1214.  As  arithmetic  and  algebra  are  sciences  of  great  clear- 
ness, certainty,  and  extent,  which  are  immediately  conversant 
about  signs,  upon  the  skilful  use  whereof  they  entirely  depend, 
so  a  little  attention  to  them  may  possibly  help  us  to  judge 
of  the  progress  of  the  mind  in  other  sciences,  which,  though 
differing  in  nature,  design,  and  object,  may  yet  agree  in  the 
general  methods  of  proof  and  inquiry. — BERKELEY,  GEORGE. 

Alciphron,  or  the  Minute  Philosopher,  Dialogue 
7,  sect.  12. 

1215.  In  general  the  position  as  regards  all  such  new  calculi 
is  this — That  one  cannot  accomplish  by  them  anything  that 
could  not  be  accomplished  without  them.    However,  the  ad- 
vantage is,  that,  provided  such  a  calculus  corresponds  to  the 


198  MEMORABILIA   MATHEMATICA 

inmost  nature  of  frequent  needs,  anyone  who  masters  it  thor- 
oughly is  able — without  the  unconscious  inspiration  of  genius 
which  no  one  can  command — to  solve  the  respective  problems, 
yea,  to  solve  them  mechanically  in  complicated  cases  ha  which, 
without  such  aid,  even  genius  becomes  powerless.  Such  is  the 
case  with  the  invention  of  general  algebra,  with  the  differential 
calculus,  and  in  a  more  limited  region  with  Lagrange's  calculus 
of  variations,  with  my  calculus  of  congruences,  and  with  Mob- 
ius's  calculus.  Such  conceptions  unite,  as  it  were,  into  an 
organic  whole  countless  problems  which  otherwise  would  remain 
isolated  and  require  for  their  separate  solution  more  or  less 
application  of  inventive  genius. — GAUSS,  C.  J. 

Werke,  Bd.  8,  p.  298. 

1216.  The  invention  of  what  we  may  call  primary  or  funda- 
mental notation  has  been  but  little  indebted  to  analogy,  evi- 
dently owing  to  the  small  extent  of  ideas  hi  which  comparison 
can  be  made  useful.    But  at  the  same  time  analogy  should  be 
attended  to,  even  if  for  no  other  reason  than  that,  by  making  the 
invention  of  notation  an  art,  the  exertion  of  individual  caprice 
ceases  to  be  allowable.    Nothing  is  more  easy  than  the  invention 
of  notation,  and  nothing  of  worse  example  and  consequence  than 
the  confusion  of  mathematical  expressions  by  unknown  symbols. 
If  new  notation  be  advisable,  permanently  or  temporarily,  it 
should  carry  with  it  some  mark  of  distinction  from  that  which  is 
already  hi  use,  unless  it  be  a  demonstrable  extension  of  the 
latter. — DE  MORGAN,  A. 

Calculus  of  Functions;  Encyclopedia  Metro- 
politana,  Addition  to  Article  26. 

1217.  Before  the  introduction  of  the  Arabic  notation,  multi- 
plication was  difficult,  and  the  division  even  of  integers  called 
into  play  the  highest  mathematical  faculties.    Probably  nothing 
hi  the  modern  world  could  have  more  astonished  a  Greek  mathe- 
matician than  to  learn  that,  under  the  influence  of  compulsory 
education,  the  whole  population  of  Western  Europe,  from  the 
highest  to  the  lowest,  could  perform  the  operation  of  division 
for  the  largest  numbers.    This  fact  would  have  seemed  to  him 
a  sheer  impossibility.  .  .  .  Our  modern  power  of  easy  reckoning 


MATHEMATICS  AS  A   LANGUAGE  199 

with  decimal  fractions  is  the  most  miraculous  result  of  a  perfect 
notation. — WHITEHEAD,  A.  N. 

Introduction    to    Mathematics    (New    York, 

1911),  p.  59. 

1218.  Mathematics  is  often  considered  a  difficult  and  myste- 
rious science,  because  of  the  numerous  symbols  which  it  em- 
ploys.    Of  course,  nothing  is  more  incomprehensible  than  a 
symbolism  which  we  do  not  understand.    Also  a  symbolism, 
which  we  only  partially  understand  and  are  unaccustomed  to 
use,  is  difficult  to  follow.    In  exactly  the  same  way  the  technical 
terms  of  any  profession  or  trade  are  incomprehensible  to  those 
who  have  never  been  trained  to  use  them.    But  this  is  not  be- 
cause they  are  difficult  in  themselves.    On  the  contrary  they 
have  invariably  been  introduced  to  make  things  easy.    So  in 
mathematics,  granted  that  we  are  giving  any  serious  attention 
to  mathematical  ideas,  the  symbolism  is  invariably  an  immense 
simplification. — WHITEHEAD,  A.  N. 

Introduction  to  Mathematics  (New  York,  1911), 
pp.  59-60. 

1219.  Symbolism  is  useful  because  it  makes  things  difficult. 
Now  in  the  beginning  everything  is  self-evident,  and  it  is  hard  to 
see  whether  one  self-evident  proposition  follows  from  another  or 
not.    Obviousness  is  always  the  enemy  to  correctness.    Hence 
we  must  invent  a  new  and  difficult  symbolism  in  which  nothing 
is  obvious.  .  .  .  Thus  the  whole  of  Arithmetic  and  Algebra  has 
been  shown  to  require  three  indefinable  notions  and  five  in- 
demonstrable propositions. — RUSSELL,  BERTRAND. 

International  Monthly,  1901,  p.  85. 

1220.  The  employment  of  mathematical  symbols  is  perfectly 
natural  when  the  relations  between  magnitudes  are  under  dis- 
cussion; and  even  if  they  are  not  rigorously  necessary,  it  would 
hardly  be  reasonable  to  reject  them,  because  they  are  not 
equally  familiar  to  all  readers  and  because  they  have  sometimes 
been  wrongly  used,  if  they  are  able  to  facilitate  the  exposition 
of  problems,  to  render  it  more  concise,  to  open  the  way  to  more 
extended  developments,  and  to  avoid  the  digressions  of  vague 
argumentation. — COURNOT,  A. 

Theory  of  Wealth  [N.  T.  Bacon],  (New  York, 
1897),  pp.  8-4. 


200  MEMORABILIA   MATHEMATICA 

1221.  An    all-inclusive    geometrical    symbolism,    such    as 
Hamilton  and  Grassmann  conceived  of,  is  impossible. 

BURKHARDT,    H. 

Jahresbericht    der    Deutschen    Mathematiker' 
Vereinigung,  Bd.  5,  p.  52. 

1222.  The  language  of  analysis,  most  perfect  of  all,  being  in 
itself  a  powerful  instrument  of  discoveries,  its  notations,  espe- 
cially when  they  are  necessary  and  happily  conceived,  are  so 
many  germs  of  new  calculi. — LAPLACE. 

Oeuvres,  t.  7  (Paris,  1896),  p.  xl. 


CHAPTER  XIII 

MATHEMATICS  AND  LOGIC 

1301.  Mathematics  belongs  to  every  inquiry,  moral  as  well 
as  physical.     Even  the  rules  of  logic,  by  which  it  is  rigidly 
bound,  could  not  be  deduced  without  its  aid.     The  laws  of 
argument  admit  of  simple  statement,  but  they  must  be  curiously 
transposed  before  they  can  be  applied  to  the  living  speech  and 
verified  by  observation.     In  its  pure  and  simple  form  the 
syllogism  cannot  be  directly  compared  with  all  experience,  or  it 
would  not  have  required  an  Aristotle  to  discover  it.     It  must  be 
transmuted  into  all  the  possible  shapes  in  which  reasoning  loves 
to  clothe  itself.    The  transmutation  is  the  mathematical  process 
in  the  establishment  of  the  law. — PEIBCE,  BENJAMIN. 

Linear  Associative  Algebra;  American  Journal 
of  Mathematics,  Vol.  4  (1881),  p.  97. 

1302.  In  mathematics  we  see  the  conscious  logical  activity  of 
our  mind  in  its  purest  and  most  perfect  form;  here  is  made 
manifest  to  us  all  the  labor  and  the  great  care  with  which  it 
progresses,  the  precision  which  is  necessary  to  determine  exactly 
the  source  of  the  established  general  theorems,  and  the  difficulty 
with  which  we  form  and  comprehend  abstract  conceptions;  but 
we  also  learn  here  to  have  confidence  in  the  certainty,  breadth, 
and  fruitfulness  of  such  intellectual  labor. — HELMHOLTZ,  H. 

Vortrage  und  Reden,  Bd.   1    (Braunschweig, 
1896),  p.  176. 

1303.  Mathematical  demonstrations  are  a  logic  of  as  much  or 
more  use,  than  that  commonly  learned  at  schools,  serving  to  a 
just  formation  of  the  mind,  enlarging  its  capacity,  and  strength- 
ening it  so  as  to  render  the  same  capable  of  exact  reasoning,  and 
discerning  truth  from  falsehood  in  all  occurrences,  even  in 
subjects  not  mathematical.     For  which  reason  it  is  said,  the 
Egyptians,  Persians,  and  Lacedaemonians  seldom  elected  any 
new  kings,  but  such  as  had  some  knowledge  in  the  mathe- 

201 


202  MEMORABILIA    MATHEMATICA 

matics,  imagining  those,  who  had  not,  men  of  imperfect  judg- 
ments, and  unfit  to  rule  and  govern. — FRANKLIN,  BENJAMIN. 

Usefulness  of  Mathematics;  Works   (Boston, 

1840),  Vol.  2,  p.  68. 

1304.  The  mathematical  conception  is,  from  its  very  nature, 
abstract;  indeed  its  abstractness  is  usually  of  a  higher  order 
than  the  abstractness  of  the  logician. — CHRYSTAL,  GEORGE. 

Encyclopedia    Britannica    (Ninth    Edition), 
Article  "Mathematics." 

1305.  Mathematics,  that  giant  pincers  of  scientific  logic  .  .  . 

HALSTED,  G.  B. 

Science  (1905),  p.  161. 

1306.  Logic  has  borrowed  the  rules  of  geometry  without 
understanding  its  power.  ...  I  am  far  from  placing  logicians 
by  the  side  of  geometers  who  teach  the  true  way  to  guide  the 
reason.  .  .  .  The  method  of  avoiding  error  is  sought  by  every 
one.    The  logicians  profess  to  lead  the  way,  the  geometers  alone 
reach  it,  and  aside  from  their  science  there  is  no  true  demonstra- 
tion.— PASCAL. 

Quoted    by    A.    Rebiere:     Mathematiques   et 
Mathematiciens  (Paris,  1898),  pp.  162-163. 

1307.  Mathematics,  like  dialectics,  is  an  organ  of  the  higher 
sense,  in  its  execution  it  is  an  art  like  eloquence.     To  both 
nothing  but  the  form  is  of  value;  neither  cares  anything  for 
content.    Whether  mathematics  considers  pennies  or  guineas, 
whether  rhetoric  defends  truth  or  error,  is  perfectly  immaterial 
to  either. — GOETHE. 

Spruche  in  Prosa,  Natur  IV,  946. 

130S.  Confined  to  its  true  domain,  mathematical  reasoning  is 
admirably  adapted  to  perform  the  universal  office  of  sound 
logic:  to  induce  in  order  to  deduce,  in  order  to  construct.  .  .  . 
It  contents  itself  to  furnish,  in  the  most  favorable  domain,  a 
model  of  clearness,  of  precision,  and  consistency,  the  close 
contemplation  of  which  is  alone  able  to  prepare  the  mind  to 
render  other  conceptions  also  as  perfect  as  their  nature  per- 
mits. Its  general  reaction,  more  negative  than  positive,  must 


MATHEMATICS   AND   LOGIC  203 

consist,  above  all,  in  inspiring  us  everywhere  with  an  invincible 
aversion  for  vagueness,  inconsistency,  and  obscurity,  which  may 
always  be  really  avoided  in  any  reasoning  whatsoever,  if  we 
make  sufficient  effort. — COMTE,  A. 

Subjective  Synthesis. 

1309.  Formal  thought,  consciously  recognized  as  such,  is  the 
means  of  all  exact  knowledge;  and  a  correct  understanding  of  the 
main  formal  sciences,  Logic  and  Mathematics,  is  the  proper  and 
only  safe  foundation  for  a  scientific  education. 

LEFEVBE,  ARTHUR. 
Number  and  its  Algebra  (Boston,  Sect.  222.) 

1310.  It  has  come  to  pass,  I  know  not  how,  that  Mathe- 
matics and  Logic,  which  ought  to  be  but  the  handmaids  of 
Physic,  nevertheless  presume  on  the  strength  of  the  certainty 
which  they  possess  to  exercise  dominion  over  it. 

BACON,  FRANCIS. 
De  Augmentis,  Bk.  3. 

1311.  We  may  regard  geometry  as  a  practical  logic,  for  the 
truths  which  it  considers,  being  the  most  simple  and  most 
sensible  of  all,  are,  for  this  reason,  the  most  susceptible  to  easy 
and  ready  application  of  the  rules  of  reasoning. — D'ALEMBERT. 

Quoted  in  A .  Rebiere:  Mathematiques  et  Mathe- 
maticiens  (Paris,  1898),  pp.  151-152. 

1312.  There  are  notable  examples  enough  of  demonstration 
outside  of  mathematics,  and  it  may  be  said  that  Aristotle  has 
already  given  some  in  his  "Prior  Analytics."    In  fact  logic  is  as 
susceptible  of  demonstration  as  geometry,  .  .  .  Archimedes  is 
the  first,  whose  works  we  have,  who  has  practised  the  art  of 
demonstration  upon  an  occasion  where  he  is  treating  of  physics, 
as  he  has  done  in  his  book  on  Equilibrium.     Furthermore, 
jurists  may  be  said  to  have  many  good  demonstrations;  espe- 
cially the  ancient  Roman  jurists,  whose  fragments  have  been 
preserved  to  us  in  the  Pandects. — LEIBNITZ,  G.  W. 

New  Essay  on  Human  Understanding  [Lang- 
ley],  Bk.  4,  chap.  2,  sect.  12. 


204  MEMORABILIA    MATHEMATICA 

1313.  It  is  commonly  considered  that  mathematics  owes  its 
certainty  to  its  reliance  on  the  immutable  principles  of  formal 
logic.    This  ...  is  only  half  the  truth  imperfectly  expressed. 
The  other  half  would  be  that  the  principles  of  formal  logic  owe 
such  a  degree  of  permanence  as  they  have  largely  to  the  fact 
that  they  have  been  tempered  by  long  and  varied  use  by  mathe- 
maticians.   "A  vicious  circle!"  you  will  perhaps  say.    I  should 
rather  describe  it  as  an  example  of  the  process  known  by  mathe- 
maticians as  the  method  of  successive  approximation. 

BOCHER,  MAXIME. 

Bulletin    of    the    American    Mathematical 
Society,  Vol.  11,  p.  120. 

1314.  Whatever  advantage  can  be  attributed  to  logic  in 
directing  and  strengthening  the  action  of  the  understanding  is 
found  in  a  higher  degree  in  mathematical  study,  with  the  im- 
mense added  advantage  of  a  determinate  subject,  distinctly  cir- 
cumscribed, admitting  of  the  utmost  precision,  and  free  from 
the  danger  which  is  inherent  in  all  abstract  logic, — of  leading  to 
useless  and  puerile  rules,  or  to  vain  ontological  speculations. 
The  positive  method,  being  everywhere  identical,  is  as  much  at 
home  in  the  art  of  reasoning  as  anywhere  else:  and  this  is  why 
no  science,  whether  biology  or  any  other,  can  offer  any  kind  of 
reasoning,  of  which  mathematics  does  not  supply  a  simpler  and 
purer  counterpart.    Thus,  we  are  enabled  to  eliminate  the  only 
remaining  portion  of  the  old  philosophy  which  could  even  ap- 
pear to  offer  any  real  utility;  the  logical  part,  the  value  of  which 
is  irrevocably  absorbed  by  mathematical  science. — COMTE,  A. 

Positive   Philosophy,    [Martineau],    (London, 
1875),  Vol.1,  pp.  321-322. 

1315.  We  know  that  mathematicians  care  no  more  for  logic 
than  logicians  for  mathematics.    The  two  eyes  of  exact  science 
are  mathematics  and  logic :  the  mathematical  sect  puts  out  the 
logical  eye,  the  logical  sect  puts  out  the  mathematical  eye;  each 
believing  that  it  can  see  better  with  one  eye  than  with  two. 

DE  MORGAN,  A. 

Quoted  in  F.  Cajori:  History  of  Mathematics 
(New  York,  1897),  p.  316. 


MATHEMATICS   AND    LOGIC  205 

1316.  The  progress  of  the  art  of  rational  discovery  depends 
in  a  great  part  upon  the  art  of  characteristic  (ars  characteristica) . 
The  reason  why  people  usually  seek  demonstrations  only  in 
numbers  and  lines  and  things  represented  by  these  is  none  other 
than  that  there  are  not,  outside  of  numbers,  convenient  char- 
acters corresponding  to  the  notions. — LEIBNITZ,  G.  W. 

Philosophische  Schriften  [Gerhardt]  Bd.  8,  p. 
198. 

1317.  The  influence  of  the  mathematics  of  Leibnitz  upon  his 
philosophy  appears  chiefly  in  connection  with  his  law  of  con- 
tinuity and  his  prolonged  efforts  to  establish  a  Logical  Cal- 
culus. ...  To  find  a  Logical  Calculus  (implying  a  universal 
philosophical  language  or  system  of  signs)  is  an  attempt  to 
apply  in  theological  and  philosophical  investigations  an  analytic 
method  analogous  to  that  which  had  proved  so  successful  in 
Geometry  and  Physics.    It  seemed  to  Leibnitz  that  if  all  the 
complex  and  apparently  disconnected  ideas  which  make  up  our 
knowledge  could  be  analysed  into  their  simple  elements,  and  if 
these  elements  could  each  be  represented  by  a  definite  sign,  we 
should  have  a  kind  of  "  alphabet  of  human  thoughts."    By  the 
combination  of  these  signs  (letters  of  the  alphabet  of  thought)  a 
system  of  true  knowledge  would  be  built  up,  in  which  reality 
would  be  more  and  more  adequately  represented   or  sym- 
bolized. ...  In  many  cases  the  analysis  may  result  in  an 
infinite  series  of  elements;  but  the  principles  of  the  Infinitesimal 
Calculus  in  mathematics  have  shown  that  this  does  not  neces- 
sarily render  calculation  impossible  or  inaccurate.     Thus  it 
seemed  to  Leibnitz  that  a  synthetic  calculus,  based  upon  a 
thorough  analysis,  would  be  the  most  effective  instrument  of 
knowledge  that  could  be  devised.    "  I  feel,"  he  says,  "  that  con- 
troversies can  never  be  finished,  nor  silence  imposed  upon  the 
Sects,  unless  we  give  up  complicated  reasonings  in  favor  of 
simple  calculations,  words  of  vague  and  uncertain  meaning  in 
favor  of  fixed  symbols."  Thus  it  will  appear  that  "  every  paralo- 
gism is  nothing  but  an  error  of  calculation' '    ' '  When  controversies 
arise,  there  will  be  no  more  necessity  of  disputation  between  two 
philosophers  than  between  two  accountants.    Nothing  will  be 
needed  but  that  they  should  take  pen  in  hand,  sit  down  with 


206  MEMORABILIA    MATHEMATICA 

their  counting-tables,  and  (having  summoned  a  friend,  if  they 
like)  say  to  one  another:  Let  us  calculate." — LATTA,  ROBERT. 

Leibnitz,  The  Monadology,  etc.  (Oxford,  1898), 

p.  85. 

1318.  Pure  mathematics  was  discovered  by  Boole  in  a  work 
which  he  called  "The  Laws  of  Thought".  .  .  .  His  work  was 
concerned  with  formal  logic,  and  this  is  the  same  thing  as 
mathematics. — RUSSELL,  BERTRAND. 

International  Monthly,  1901,  p.  88. 

1319.  Mathematics  is  but  the  higher  development  of  Sym- 
bolic Logic. — WHETHAM,  W.  C.  D. 

Recent    Development    of    Physical    Science 
(Philadelphia,  1904),  P-  34. 

1320.  Symbolic  Logic  has  been  disowned  by  many  logicians 
on  the  plea  that  its  interest  is  mathematical,  and  by  many 
mathematicians  on  the  plea  that  its  interest  is  logical. 

WHITEHEAD,  A.  N. 

Universal  Algebra  (Cambridge,  1898),  Preface, 
p.  6. 

1321.  .  .  .  the  two  great  components  of  the  critical  move- 
ment, though  distinct  in  origin  and  following  separate  paths, 
are  found  to  converge  at  last  in  the  thesis:  Symbolic  Logic  is 
Mathematics,  Mathematics  is  Symbolic  Logic,  the  twain  are  one. 

KEYSER,  C.  J. 

Lectures    on    Science,    Philosophy    and    Art 
(New  York,  1908),  p.  19. 

1322.  The  emancipation  of  logic  from  the  yoke  of  Aristotle 
very  much  resembles  the  emancipation  of  geometry  from  the 
bondage  of  Euclid;  and,  by  its  subsequent  growth  and  diversifi- 
cation, logic,  less  abundantly  perhaps  but  not  less  certainly  than 
geometry,  has  illustrated  the  blessings  of  freedom. 

KEYSER,  C.  J. 
Science,  Vol.  85  (1912),  p.  108. 

1323.  I  would  express  it  as  my  personal  view,  which  is  proba- 
bly not  yet  shared  generally,  that  pure  mathematics  seems  to 


MATHEMATICS   AND   LOGIC  207 

me  merely  a  branch  of  general  logic;  that  branch  which  is  based 
on  the  concept  of  numbers,  to  whose  economic  advantages  is  to 
be  attributed  the  tremendous  development  which  this  particular 
branch  has  undergone  as  compared  with  the  remaining  branches 
of  logic,  which  until  the  most  recent  times  have  remained  al- 
most stationary. — SCHRODER,  E. 

Ueber  Pasigraphie  etc.;  Verhandlungen  des  1 .  In- 
ternationalen  Mathematiker-Kongresses  (Leipzig, 
1898),  p.  149. 

1324.  If  logical  training  is  to  consist,  not  in  repeating  bar- 
barous scholastic  formulas  or  mechanically  tacking  together 
empty  majors  and  minors,  but  in  acquiring  dexterity  in  the  use 
of  trustworthy  methods  of  advancing  from  the  known  to  the 
unknown,  then  mathematical  investigation  must  ever  remain 
one  of  its  most  indispensable  instruments.    Once  inured  to  the 
habit  of  accurately  imagining  abstract  relations,  recognizing  the 
true  value  of  symbolic  conceptions,  and  familiarized  with  a 
fixed  standard  of  proof,  the  mind  is  equipped  for  the  considera- 
tion of  quite  other  objects  than  lines  and  angles.     The  twin 
treatises  of  Adam  Smith  on  social  science,  wherein,  by  deducing 
all  human  phenomena  first  from  the  unchecked  action  of  selfish- 
ness and  then  from  the  unchecked  action  of  sympathy,  he 
arrives  at  mutually-limiting  conclusions  of  transcendent  practi- 
cal importance,  furnish  for  all  time  a  brilliant  illustration  of  the 
value  of  mathematical  methods  and  mathematical  discipline. 

FISKE,  JOHN. 

Darwinism  and  other  Essays  (Boston,  1893), 
pp.  297-298. 

1325.  No  irrational  exaggeration  of  the  claims  of  Mathe- 
matics can  ever  deprive  that  part  of  philosophy  of  the  property 
of  being  the  natural  basis  of  all  logical  education,  through  its 
simplicity,   abstractness,   generality,   and  freedom   from   dis- 
turbance by  human  passion.    There,  and  there  alone,  we  find  in 
full  development  the  art  of  reasoning,  all  the  resources  of 
which,  from  the  most  spontaneous  to  the  most  sublime,  are 
continually  applied  with  far  more  variety  and  fruitfulness  than 
elsewhere;  .  .  .  The  more   abstract  portion  of  mathematics 


208  MEMORABILIA   MATHEMATICA 

may  in  fact  be  regarded  as  an  immense  repository  of  logical 
resources,  ready  for  use  in  scientific  deduction  and  co-ordination. 

COMTE,  A. 

Positive    Philosphy    [Martineau],     (London, 
1875),  Vol.  2,  p.  439. 

1326.  Logic  it  is  called  [referring  to  Whitehead  and  Russell's 
Principia  Mathematical  and  logic  it  is,  the  logic  of  propositions 
and  functions  and  classes  and  relations,  by  far  the  greatest  (not 
merely  the  biggest)  logic  that  our  planet  has  produced,  so  much 
that  is  new  in  matter  and  in  manner;  but  it  is  also  mathematics, 
a  prolegomenon  to  the  science,  yet  itself  mathematics  in  its 
most  genuine  sense,  differing  from  other  parts  of  the  science 
only  in  the  respects  that  it  surpasses  these  in  fundamentality, 
generality  and  precision,  and  lacks  traditionality.  Few  will 
read  it,  but  all  will  feel  its  effect,  for  behind  it  is  the  urgence  and 
push  of  a  magnificent  past:  two  thousand  five  hundred  years  of 
record  and  yet  longer  tradition  of  human  endeavor  to  think 

aright. — KEYSER,  C.  J. 

Science,  Vol.  35  (1912),  p.  110. 


CHAPTER  XIV 

MATHEMATICS   AND    PHILOSOPHY 

1401.  Socrates  is  praised  by  all  the  centuries  for  having  called 
philosophy  from  heaven  to  men  on  earth;  but  if,  knowing  the 
condition  of  our  science,  he  should  come  again  and  should  look 
once  more  to  heaven  for  a  means  of  curing  men,  he  would  there 
find  that  to  mathematics,  rather  than  to  the  philosophy  of 
today,  had  been  given  the  crown  because  of  its  industry  and  its 
most  happy  and  brilliant  successes. — HERBART,  J.  F. 

Werke  [Kehrbach],  (Langensalza,  1890),  Bd.  5, 
p.  95. 

1402.  It  is  the  embarrassment  of  metaphysics  that  it  is  able 
to  accomplish  so  little  with  the  many  things  that  mathematics 
offers  her. — KANT,  E. 

Metaphysische  Anfangsgriinde  der  Naturwis- 
senschaft,  Vorrede. 

1403.  Philosophers,  when  they  have  possessed  a  thorough 
knowledge  of  mathematics,  have  been  among  those  who  have 
enriched  the  science  with  some  of  its  best  ideas.    On  the  other 
hand  it  must  be  said  that,  with  hardly  an  exception,  all  the 
remarks  on  mathematics  made  by  those  philosophers  who 
have  possessed  but  a  slight  or  hasty  or  late-acquired  knowledge 
of  it  are  entirely  worthless,  being  either  trivial  or  wrong. 

WHITEHEAD,  A.  N. 

Introduction    to    Mathematics    (New    York, 
1911),  p.  113. 

1404.  The  union  of  philosophical  and  mathematical  produc- 
tivity, which  besides  in  Plato  we  find  only  in  Pythagoras, 
Descartes  and  Leibnitz,  has  always  yielded  the  choicest  fruits  to 
mathematics:  To  the  first  we  owe  scientific  mathematics  in 
general,  Plato  discovered  the  analytic  method,  by  means  of 
which  mathematics  was  elevated  above  the  view-point  of  the 
elements,  Descartes  created  the  analytical  geometry,  our  own 

209 


210  MEMORABILIA   MATHEMATICA 

illustrious  countryman  discovered  the  infinitesimal  calculus — 
and  just  these  are  the  four  greatest  steps  in  the  development  of 
mathematics. — HANKEL,  HERMANN. 

Geschichte  der  Mathematik  im  Altertum  und 
im  MittelaUer  (Leipzig,  1874),  PP-  149-150. 

1405.  Without  mathematics  one  cannot  fathom  the  depths 
of  philosophy;  without  philosophy  one   cannot  fathom  the 
depths  of  mathematics;  without  the  two  one  cannot  fathom 
anything. — BORDAS-DEMOULINS. 

Quoted  in  A.  Rebie're:  Mathematiques  et  Mathe- 
maticiens  (Paris,  1898),  p.  147. 

1406.  In  the  end  mathematics  is  but  simple  philosophy,  and 
philosophy,  higher  mathematics  in  general. — NOVALIS. 

Schriften  (Berlin,  1901),  Teil  2,  p.  443. 

1407.  It  is  a  safe  rule  to  apply  that,  when  a  mathematical  or 
philosophical  author  writes  with  a  misty  profundity,  he  is 
talking  nonsense. — WHITEHEAD,  A.  N. 

Introduction  to  Mathematics  (New  York,  1911), 
p.  227. 

1408.  The  real  finisher  of  our  education  is  philosophy,  but  it  is 
the  office  of  mathematics  to  ward  off  the  dangers  of  philosophy. 

HERBART,  J.  F. 

Pestalozzi's  Idee  eines  ABC  der  Anschauung; 
Werke  [Kehrbach],  (Langensalza,  1890), 
Bd.  1,  p.  168. 

1409.  Since  antiquity  mathematics  has  been  regarded  as  the 
most  indispensable  school  for  philosophic  thought  and  in  its 
highest  spheres  the  research  of  the  mathematician  is  indeed  most 
closely  related  to  pure  speculation.     Mathematics  is  the  most 
perfect  union  between  exact  knowledge  and  theoretical  thought. 

CURTIUS,  E. 
Berliner  Monatsberichte  (1873),  p.  617. 

1410.  Geometry  has  been,  throughout,  of  supreme  importance 
in  the  history  of  knowledge. — RUSSELL,  BERTRAND. 

Foundations  of  Geometry  (Cambridge,  1897), 
P-  64. 


MATHEMATICS   AND    PHILOSOPHY  211 

1411.  He  is  unworthy  of  the  name  of  man  who  is  ignorant  of 
the  fact  that  the  diagonal  of  a  square  is  incommensurable  with 

its  side. — PLATO. 

Quoted  by  Sophie  Germain:  Memoire  sur  les 
surfaces  elastiques. 

1412.  Mathematics,  considered  as  a  science,  owes  its  origin  to 
the  idealistic  needs  of  the  Greek  philosophers,  and  not  as  fable 
has  it,  to  the  practical  demands  of  Egyptian  economics.  .  .  . 
Adam  was  no  zoologist  when  he  gave  names  to  the  beasts  of  the 
field,  nor  were  the  Egyptian  surveyors  mathematicians. 

HANKEL,  H. 

Die  Entwickelung  der  Mathematik  in  den 
letzten  Jahrhunderten  (Tubingen,  1884),  P-  7. 

1413.  There  are  only  two  ways  open  to  man  for  attaining  a 
certain  knowledge  of  truth :  clear  intuition  and  necessary  deduc- 
tion.— DESCARTES. 

Rules  for  the  Direction  of  the  Mind;  Torrey's 
The  Philosophy  of  Descartes  (New  York, 
1892),  p.  104. 

1414.  Mathematicians  have,  in  many  cases,  proved  some 
things  to  be  possible  and  others  to  be  impossible,  which,  without 
demonstration,   would  not  have  been  believed  .  .  .  Mathe- 
matics afford  many  instances  of  impossibilities  in  the  nature  of 
things,  which  no  man  would  have  believed,  if  they  had  not  been 
strictly  demonstrated.     Perhaps,  if  we  were  able  to  reason 
demonstratively  in  other  subjects,  to  as  great  extent  as  in 
mathematics,  we  might  find  many  things  to  be  impossible, 
which  we  conclude,  without  hesitation,  to  be  possible. 

REID,  THOMAS. 

Essay  on  the  Intellectual  Powers  of  Man, 
Essay  4,  chap.  3. 

1415.  If  philosophers  understood  mathematics,  they  would 
know  that  indefinite  speech,  which  permits  each  one  to  think 
what  he  pleases  and  produces  a  constantly  increasing  difference 
of  opinion,  is  utterly  unable,  in  spite  of  all  fine  words  and  even 
in  spite  of  the  magnitude  of  the  objects  which  are  under  con- 
templation, to  maintain  a  balance  against  a  science  which  in- 


212  MEMORABILIA   MATHEMATICA 

structs  and  advances  through  every  word  which  it  utters  and 
which  at  the  same  time  wins  for  itself  endless  astonishment,  not 
through  its  survey  of  immense  spaces,  but  through  the  exhibi- 
tion of  the  most  prodigious  human  ingenuity  which  surpasses  all 
power  of  description. — HERBART,  J.  F. 

Werke  Kehrbach  (Langensalza,  1890),  Bd.  5, 
p.  105. 

1416.  German  intellect  is  an  excellent  thing,  but  when  a 
German  product  is  presented  it  must  be  analysed.     Most 
probably  it  is  a  combination  of  intellect  (I)  and  tobacco-smoke 
(T).     Certainly  IsTi,  and  I^Ti,  occur;  but  IiT3  is  more  com- 
mon, and  l2Tis  and  IiT2o  occur.    In  many  cases  metaphysics  (M) 
occurs  and  I  hold  that  IftTbMc  never  occurs  without  b  +  c  >  2a. 

N.  B. — Be  careful,  hi  analysing  the  compounds  of  the  three, 
not  to  confound  T  and  M,  which  are  strongly  suspected  to  be 
isomorphic.  Thus,  IiT3M3  may  easily  be  confounded  with 
IiT6.  As  far  as  I  dare  say  anything,  those  who  have  placed 
Hegel,  Fichte,  etc.,  in  the  rank  of  the  extenders  of  Kant  have 
imagined  T  and  M  to  be  identical. — DE  MORGAN,  A. 

Graves'  Life  of  W.  R.  Hamilton  (New  York, 
1882-1889],  Vol.  13,  p.  446. 

1417.  The  discovery  [of  Ceres]  was  made  by  G.  Piazzi  of 
Palermo;  and  it  was  the  more  interesting  as  its  announcement 
occurred  simultaneously  with  a  publication  by  Hegel  in  which 
he  severely  criticized  astronomers  for  not  paying  more  attention 
to  philosophy,  a  science,  said  he,  which  would  at  once  have 
shown  them  that  there  could  not  possibly  be  more  than  seven 
planets,  and  a  study  of  which  would  therefore  have  prevented  an 
absurd  waste  of  time  hi  looking  for  what  hi  the  nature  of  things 
could  never  be  found. — BALL,  W.  W.  R. 

History  of  Mathematics  (London,  1901),  p.  458. 

1418.  But  who  shall  parcel  out 

His  intellect  by  geometric  rules, 

Split  like  a  province  into  round  and  square? 

WORDSWORTH. 
The  Prelude,  Bk.  2. 


MATHEMATICS   AND   PHILOSOPHY  213 

1419.  And  Proposition,  gentle  maid, 

Who  soothly  ask'd  stern  Demonstration's  aid,  .  .  . 

COLERIDGE,  S.  T. 
A  Mathematical  Problem. 

1420.  Mathematics  connect  themselves  on  the  one  side  with 
common  life  and  physical  science;  on  the  other  side  with  philoso- 
phy in  regard  to  our  notions  of  space  and  time,  and  in  the  ques- 
tions which  have  arisen  as  to  the  universality  and  necessity  of 
the  truths  of  mathematics  and  the  foundation  of  our  knowledge 
of  them. — CAYLEY,  ARTHUR. 

British  Association  Address  (1883);  Collected 
Mathematical  Papers,  Vol.  11,  p.  430. 

1427.  Mathematical  teaching  .  .  .  trains  the  mind  to  capaci- 
ties, which  .  .  .  are  of  the  closest  kin  to  those  of  the  greatest 
metaphysician  and  philosopher.  There  is  some  color  of  truth 
for  the  opposite  doctrine  in  the  case  of  elementary  algebra. 
The  resolution  of  a  common  equation  can  be  reduced  to  almost 
as  mechanical  a  process  as  the  working  of  a  sum  in  arithmetic. 
The  reduction  of  the  question  to  an  equation,  however,  is  no 
mechanical  operation,  but  one  which,  according  to  the  degree  of 
its  difficulty,  requires  nearly  every  possible  grade  of  ingenuity: 
not  to  speak  of  the  new,  and  hi  the  present  state  of  the  science 
insoluble,  equations,  which  start  up  at  every  fresh  step  at- 
tempted in  the  application  of  mathematics  to  other  branches  of 
knowledge. — MILL,  J.  S. 

An  Examination  of  Sir  William  Hamilton's 
Philosophy  (London,  1878),  p.  615. 

1422.  The  value  of  mathematical  instruction  as  a  preparation 
for  those  more  difficult  investigations,  consists  hi  the  applica- 
bility not  of  its  doctrines,  but  of  its  methods.  Mathematics 
will  ever  remain  the  most  perfect  type  of  the  Deductive  Method 
in  general;  and  the  applications  of  mathematics  to  the  simpler 
branches  of  physics,  furnish  the  only  school  in  which  philoso- 
phers can  effectually  learn  the  most  difficult  and  important 
portion  of  their  art,  the  employment  of  the  laws  of  the  simpler 
phenomena  for  explaining  and  predicting  those  of  the  more 
complex.  These  grounds  are  quite  sufficient  for  deeming  mathe- 


214  MEMORABILIA   MATHEMATICA 

matical  training  an  indispensable  basis  of  real  scientific  educa- 
tion, and  regarding,  with  Plato,  one  who  is  ayeajfteTprfTo*;,  as 
wanting  in  one  of  the  most  essential  qualifications  for  the 
successful  cultivation  of  the  higher  branches  of  philosophy. 

MILL,  J.  S. 
System  of  Logic,  Bk.  3,  chap.  24,  sect.  9. 

1423.  In  metaphysical  reasoning,  the  process  is  always  short. 
The  conclusion  is  but  a  step  or  two,  seldom  more,  from  the  first 
principles  or  axioms  on  which  it  is  grounded,  and  the  different 
conclusions  depend  not  one  upon  another. 

It  is  otherwise  in  mathematical  reasoning.  Here  the  field  has 
no  limits.  One  proposition  leads  on  to  another,  that  to  a  third, 
and  so  on  without  end.  If  it  should  be  asked,  why  demonstra- 
tive reasoning  has  so  wide  a  field  hi  mathematics,  while,  hi  other 
abstract  subjects,  it  is  confined  within  very  narrow  limits,  I 
conceive  this  is  chiefly  owing  to  the  nature  of  quantity,  .  .  . 
mathematical  quantities  being  made  up  of  parts  without  num- 
ber, can  touch  in  innumerable  points,  and  be  compared  in 
innumerable  different  ways. — REID,  THOMAS. 

Essays  on  the  Powers  of  the  Human  Mind 
(Edinburgh,  1812},  Vol.  2,  pp.  422-423. 

1424.  The  power  of  Reason  ...  is  unquestionably  the  most 
important  by  far  of  those  which  are  comprehended  under  the 
general  title  of  Intellectual.    It  is  on  the  right  use  of  this  power 
that  our  success  in  the  pursuit  of  both  knowledge  and  of  happi- 
ness depends;  and  it  is  by  the  exclusive  possession  of  it  that  man 
is  distinguished,  in  the  most  essential  respects,  from  the  lower 
animals.    It  is,  indeed,  from  their  subserviency  to  its  operations, 
that  the  other  faculties  .  .  .  derive  their  chief  value. 

STEWART,  DUGALD. 

Philosophy  of  the  Human  Mind;  Collected 
Works  (Edinburgh,  1854),  Vol.  8,  p.  5. 

1425.  When  ...  I  asked  myself  why  was  it  then  that  the 
earliest  philosophers  would  admit  to  the  study  of  wisdom  only 
those  who  had  studied  mathematics,  as  if  this  science  was  the 
easiest  of  all  and  the  one  most  necessary  for  preparing  and 
disciplining  the  mind  to  comprehend  the  more  advanced,  I 


MATHEMATICS   AND    PHILOSOPHY  215 

suspected  that  they  had  knowledge  of  a  mathematical  science 
different  from  that  of  our  time.  .  .  . 

I  believe  I  find  some  traces  of  these  true  mathematics  in 
Pappus  and  Diophantus,  who,  although  they  were  not  of 
extreme  antiquity,  lived  nevertheless  in  times  long  preceding 
ours.  But  I  willingly  believe  that  these  writers  themselves,  by  a 
culpable  ruse,  suppressed  the  knowledge  of  them;  like  some 
artisans  who  conceal  their  secret,  they  feared,  perhaps,  that  the 
ease  and  simplicity  of  their  method,  if  become  popular,  would 
diminish  its  importance,  and  they  preferred  to  make  themselves 
admired  by  leaving  to  us,  as  the  product  of  their  art,  certain 
barren  truths  deduced  with  subtlety,  rather  than  to  teach  us 
that  art  itself,  the  knowledge  of  which  would  end  our  admiration. 

DESCARTES. 

Rules  for  the  Direction  of  the  Mind;  Philosophy 
of  Descartes  [Torrey],  (New  York,  1892},  pp. 
70-71. 

1426.  If  we  rightly  adhere  to  our  rule  [that  is,  that  we  should 
occupy  ourselves  only  with  those  subjects  in  reference  to  which 
the  mind  is  capable  of  acquiring  certain  and  indubitable  knowl- 
edge] there  will  remain  but  few  things  to  the  study  of  which  we 
can  devote  ourselves.     There  exists  in  the  sciences  hardly  a 
single  question  upon  which  men  of  intellectual  ability  have  not 
held  different  opinions.    But  whenever  two  men  pass  contrary 
judgment  on  the  same  thing,  it  is  certain  that  one  of  the  two  is 
wrong.     More  than  that,  neither  of  them  has  the  truth;  for 
if  one  of  them  had  a  clear  and  precise  insight  into  it,  he  could  so 
exhibit  it  to  his  opponent  as  to  end  the  discussion  by  compelling 
his  conviction.  ...  It  follows  from  this,  if  we  reckon  rightly, 
that  among  existing  sciences  there  remain  only  geometry  and 
arithmetic,  to  which  the  observance  of  our  rule  would  bring  us. 

DESCARTES. 

Rules  for  the  Direction  of  the  Mind;  Philos- 
ophy of  Descartes  [Torrey],  (New  York,  1892}, 
p.  62. 

1427.  The  same  reason  which  led  Plato  to  recommend  the 
study  of  arithmetic  led  him  to  recommend  also  the  study  of 
geometry.    The  vulgar  crowd  of  geometricians,  he  says,  will  not 


216  MEMORABILIA   MATHEMATICA 

understand  him.  They  have  practice  always  in  view.  They  do 
not  know  that  the  real  use  of  the  science  is  to  lead  men  to  the 
knowledge  of  abstract,  essential,  eternal  truth.  (Plato's  Repub- 
lic, Book  7).  Indeed  if  we  are  to  believe  Plutarch,  Plato  carried 
his  feeling  so  far  that  he  considered  geometry  as  degraded  by 
being  applied  to  any  purpose  of  vulgar  utility.  Archytas,  it 
seems,  had  framed  machines  of  extraordinary  power  on  mathe- 
matical principles.  (Plutarch,  Sympos.,  VIII.,  and  Life  of 
Marcellus.  The  machines  of  Archytas  are  also  mentioned  by 
Aulus  Gellius  and  Diogenes  Laertius).  Plato  remonstrated 
with  his  friend,  and  declared  that  this  was  to  degrade  a  noble 
intellectual  exercise  into  a  low  craft,  fit  only  for  carpenters  and 
wheelwrights.  The  office  of  geometry,  he  said,  was  to  discipline 
the  mind,  not  to  minister  to  the  base  wants  of  the  body.  His 
interference  was  successful;  and  from  that  time  according  to 
Plutarch,  the  science  of  mechanics  was  considered  unworthy  of 
the  attention  of  a  philosopher. — MACAULAY. 

Lord  Bacon;  Edinburgh  Review,  July,  1837. 

1428.  The  intellectual  habits  of  the  Mathematicians  are,  in 
some  respects,  the  same  with  those  [of  the  Metaphysicians]  we 
have  been  now  considering;  but,  in  other  respects,  they  differ 
widely.  Both  are  favourable  to  the  improvement  of  the  power 
of  attention,  but  not  in  the  same  manner,  nor  hi  the  same  degree. 

Those  of  the  metaphysician  give  capacity  of  fixing  the  atten- 
tion on  the  subjects  of  our  consciousness,  without  being  dis- 
tracted by  things  external;  but  they  afford  little  or  no  exercise 
to  that  species  of  attention  which  enables  us  to  follow  long 
processes  of  reasoning,  and  to  keep  hi  view  all  the  various  steps 
of  an  investigation  till  we  arrive  at  the  conclusion.  In  mathe- 
matics, such  processes  are  much  longer  than  in  any  other 
science;  and  hence  the  study  of  it  is  peculiarly  calculated  to 
strengthen  the  power  of  steady  and  concatenated  thinking, — 
a  power  which,  in  all  the  pursuits  of  life,  whether  speculative  or 
active,  is  one  of  the  most  valuable  endowments  we  can  possess. 
This  command  of  attention,  however,  it  may  be  proper  to  add, 
is  to  be  acquired,  not  by  the  practice  of  modern  methods,  but 
by  the  study  of  Greek  geometry,  more  particularly,  by  accustom- 
ing ourselves  to  pursue  long  trains  of  demonstration,  without 


MATHEMATICS   AND    PHILOSOPHY  217 

availing  ourselves  of  the  aid  of  any  sensible  diagrams;  the 
thoughts  being  directed  solely  by  those  ideal  delineations  which 
the  powers  of  conception  and  of  memory  enable  us  to  form. 

STEWART,  DUGALD. 

Philosophy  of  the  Human  Mind,  Part  3,  chap.  1 , 

sect.  8. 

1429.  They  [the  Greeks]  speculated  and  theorized  under  a 
lively  persuasion  that  a  Science  of  every  part  of  nature  was 
possible,  and  was  a  fit  object  for  the  exercise  of  a  man's  best 
faculties;  and  they  were  speedily  led  to  the  conviction  that 
such  a  science  must  clothe  its  conclusions  in  the  language  of 
mathematics.    This  conviction  is  eminently  conspicuous  in  the 
writings  of  Plato.  .  .  .  Probably  no  succeeding  step  in  the 
discovery  of  the  Laws  of  Nature  was  of  so  much  importance  as 
the  full  adoption  of  this  pervading  conviction,  that  there  must 
be  Mathematical  Laws  of  Nature,  and  that  it  is  the  business  of 
Philosophy  to  discover  these  Laws.    This  conviction  continues, 
through  all  the  succeeding  ages  of  the  history  of  the  science,  to 
be  the  animating  and  supporting  principle  of  scientific  investiga- 
tion and  discovery. — WHEWELL,  W. 

History  of  the  Inductive  Sciences,  Vol.  1,  bk.  2, 
chap.  3. 

1430.  For  to  pass  by  those  Ancients,  the  wonderful  Pythagoras, 
the  sagacious  Democritus,  the  divine  Plato,  the  most  subtle  and 
very  learned  Aristotle,  Men  whom  every  Age  has  hitherto 
acknowledged  as  deservedly  honored,  as  the  greatest  Philoso- 
phers, the  Ring-leaders  of  Arts;  in  whose  Judgments  how  much 
these  Studies  [mathematics]    were   esteemed,    is   abundantly 
proclaimed  in  History  and  confirmed  by  their  famous  Monu- 
ments, which  are  everywhere  interspersed  and  bespangled  with 
Mathematical  Reasonings  and  Examples,  as  with  so  many 
Stars;  and  consequently  anyone  not  in  some  Degree  conversant 
in  these  Studies  will  hi  vain  expect  to  understand,  or  unlock 
their  hidden  Meanings,  without  the  Help  of  a  Mathematical 
Key:  For  who  can  play  well  on  Aristotle's  Instrument  but  with  a 
Mathematical  Quill;  or  not  be  altogether  deaf  to  the  Lessons  of 
natural  Philosophy,  while  ignorant  of  Geometry?    Who  void  of 
(Geometry  shall  I  say,  or)  Arithmetic  can  comprehend  Plato's 


218  MEMORABILIA   MATHEMATICA 

Socrates  lisping  with  Children  concerning  Square  Numbers;  or 
can  conceive  Plato  himself  treating  not  only  of  the  Universe, 
but  the  Polity  of  Commonwealths  regulated  by  the  Laws  of 
Geometry,  and  formed  according  to  a  Mathematical  Plan? 

BARROW,  ISAAC. 

Mathematical     Lectures      (London,      1784), 

pp.  26-27. 

1431.  And  Reason  now  through  number,  time,  and  space 
Darts  the  keen  lustre  of  her  serious  eye; 

And  learns  from  facts  compar'd  the  laws  to  trace 
Whose  long  procession  leads  to  Deity. 

BEATTIE,  JAMES. 
The    Minstrel,    Bk.    2,    stanza   47. 

1432.  That  Egyptian  and  Chaldean  wisdom  mathematical 
wherewith  Moses  and  Daniel  were  furnished,  .  .  . 

HOOKER,  RICHARD. 

Ecclesiastical  Polity,   Bk.   8,   sect.   8. 

1433.  General  and  certain  truths  are  only  founded  in  the 
habitudes  and  relations  of  abstract  ideas.     A  sagacious  and 
methodical  application  of  our  thoughts,  for  the  finding  out  of 
these  relations,  is  the  only  way  to  discover  all  that  can  be  put 
with  truth  and  certainty  concerning  them  into  general  proposi- 
tions.   By  what  steps  we  are  to  proceed  in  these,  is  to  be  learned 
in  the  schools  of  mathematicians,  who,  from  very  plain  and 
easy  beginnings,  by  gentle  degrees,  and  a  continued  chain  of 
reasonings,  proceed  to  the  discovery  and  demonstration  of 
truths  that  appear  at  first  sight  beyond  human  capacity.    The 
art  of  finding  proofs,  and  the  admirable  method  they  have 
invented  for  the  singling  out  and  laying  hi  order  those  inter- 
mediate  ideas   that   demonstratively   show   the   equality   or 
inequality  of  unapplicable  quantities,  is  that  which  has  carried 
them  so  far  and  produced  such  wonderful  and  unexpected  dis- 
coveries; but  whether  something  like  this,  in  respect  of  other 
ideas,  as  well  as  those  of  magnitude,  may  not  in  time  be  found 
out,  I  will  not  determine.    This,  I  think,  I  may  say,  that  if 
other  ideas  that  are  the  real  as  well  as  the  nominal  essences  of 
their  species,  were  pursued  hi  the  way  familiar  to  mathemati- 


MATHEMATICS   AND    PHILOSOPHY  219 

cians,  they  would  carry  our  thoughts  further,  and  with  greater 
evidence  and  clearness  than  possibly  we  are  apt  to  imagine. 

LOCKE,  JOHN. 

An  Essay  concerning  Human  Understanding, 
Bk.  4,  chap.  12,  sect.  7. 

1434.  Those  long  chains  of  reasoning,  quite  simple  and  easy, 
which  geometers  are  wont  to  employ  in  the  accomplishment  of 
their  most  difficult  demonstrations,  led  me  to  think  that  every- 
thing which  might  fall  under  the  cognizance  of  the  human  mind 
might  be  connected  together  in  a  similar  manner,  and  that, 
provided  only  that  one  should  take  care  not  to  receive  anything 
as  true  which  was  not  so,  and  if  one  were  always  careful  to 
preserve  the  order  necessary  for  deducing  one  truth  from  an- 
other, there  would  be  none  so  remote  at  which  he  might  not  at 
last  arrive,  nor  so  concealed  which  he  might  not  discover. 

DESCARTES. 

Discourse  upon  Method,  part  2;  The  Philos- 
ophy of  Descartes  [Torrey],  (New  York,  1892), 
p.  47. 

1435.  If  anyone  wished  to  write  in  mathematical  fashion  in 
metaphysics  or  ethics,  nothing  would  prevent  him  from  so  doing 
with  vigor.    Some  have  professed  to  do  this,  and  we  have  a 
promise  of  mathematical  demonstrations  outside  of  mathe- 
matics; but  it  is  very  rare  that  they  have  been  successful.    This 
is,  I  believe,  because  they  are  disgusted  with  the  trouble  it  is 
necessary  to  take  for  a  small  number  of  readers  where  they  would 
ask  as  in  Persius:  Quis  leget  haec,  and  reply:  Vel  duo  vel  nemo. 

LEIBNITZ. 

New  Essay  concerning  Human  Understanding, 
Langley,  Bk  2,  chap.  29,  sect.  12. 

1436.  It  is  commonly  asserted  that  mathematics  and  philoso- 
phy differ  from  one  another  according  to  their  objects,  the 
former  treating  of  quantity,  the  latter  of  quality.    All  this  is 
false.    The  difference  between  these  sciences  cannot  depend  on 
their  object;  for  philosophy  applies  to  everything,  hence  also 
to  quanta,  and  so  does  mathematics  in  part,  inasmuch  as  every- 
thing has  magnitude.    It  is  only  the  different  kind  of  rational 
knowledge  or  application  of  reason  in  mathematics  and  philoso- 
phy which  constitutes  the  specific  difference  between  these  two 


220  MEMORABILIA  MATHEMATICA 

sciences.  For  philosophy  is  rational  knowledge  from  mere  con- 
cepts, mathematics,  on  the  contrary,  is  rational  knowledge  from 
the  construction  of  concepts. 

We  construct  concepts  when  we  represent  them  in  intuition 
a  priori,  without  experience,  or  when  we  represent  in  intuition 
the  object  which  corresponds  to  our  concept  of  it. — The  mathe- 
matician can  never  apply  his  reason  to  mere  concepts,  nor  the 
philosopher  to  the  construction  of  concepts. — In  mathematics 
the  reason  is  employed  in  concreto,  however,  the  intuition  is  not 
empirical,  but  the  object  of  contemplation  is  something  a  priori. 

In  this,  as  we  see,  mathematics  has  an  advantage  over  philoso- 
phy, the  knowledge  in  the  former  being  intuitive,  in  the  latter, 
on  the  contrary,  only  discursive.  But  the  reason  why  in  mathe- 
matics we  deal  more  with  quantity  lies  in  this,  that  magnitudes 
can  be  constructed  in  intuition  a  priori,  while  qualities,  on  the 
contrary,  do  not  permit  of  being  represented  in  intuition. 

KANT,  E. 

Logik;  Werke  [Hartenstein],  (Leipzig,  1868), 
Bd.  8,  pp.  23-24. 

1437.  Kant  has  divided  human  ideas  into  the  two  categories  of 
quantity  and  quality,  which,  if  true,  would  destroy  the  univer- 
sality of  Mathematics;  but  Descartes'  fundamental  conception 
of  the  relation  of  the  concrete  to  the  abstract  in  Mathematics 
abolishes  this  division,  and  proves  that  all  ideas  of  quality  are 
reducible  to  ideas  of  quantity.  He  had  in  view  geometrical 
phenomena  only;  but  his  successors  have  included  in  this 
generalization,  first,  mechanical  phenomena,  and,  more  re- 
cently, those  of  heat.  There  are  now  no  geometers  who  do  not 
consider  it  of  universal  application,  and  admit  that  every 
phenomenon  may  be  as  logically  capable  of  being  represented  by 
an  equation  as  a  curve  or  a  motion,  if  only  we  were  always 
capable  (which  we  are  very  far  from  being)  of  first  discovering, 
and  then  resolving  it. 

The  limitations  of  Mathematical  science  are  not,  then,  in  its 
nature.  The  limitations  are  in  our  intelligence:  and  by  these 
we  find  the  domain  of  the  science  remarkably  restricted,  in 
proportion  as  phenomena,  in  becoming  special,  become  com- 
plex.— COMTE,  A.  ..  ,,,-..  i  D7 

Positive  Philosophy  [Martmeau],  Bk.  1,  chap.  1. 


MATHEMATICS   AND    PHILOSOPHY  221 

1438.  The  great  advantage  of  the  mathematical  sciences  over 
the  moral  consists  in  this,  that  the  ideas  of  the  former,  being 
sensible,  are  always  clear  and  determinate,  the  smallest  distinc- 
tion between  them  being  immediately  perceptible,  and  the  same 
terms  are  still  expressive  of  the  same  ideas,  without  ambiguity 
or  variation.    An  oval  is  never  mistaken  for  a  circle,  nor  an 
hyperbola  for  an  ellipsis.     The  isosceles  and  scalenum  are 
distinguished  by  boundaries  more  exact  than  vice  and  virtue, 
right  or  wrong.    If  any  term  be  denned  in  geometry,  the  mind 
readily,  of  itself,  substitutes  on  all  occasions,  the  definition  for 
the  thing  defined:  Or  even  when  no  definition  is  employed,  the 
object  itself  may  be  represented  to  the  senses,  and  by  that 
means  be  steadily  and  clearly  apprehended.     But  the  finer 
sentiments  of  the  mind,  the  operations  of  the  understanding,  the 
various  agitations  of  the  passions,  though  really  in  themselves 
distinct,  easily  escape  us,  when  surveyed  by  reflection;  nor  is 
it  in  our  power  to  recall  the  original  object,  so  often  as  we  have 
occasion  to  contemplate  it.     Ambiguity,  by  this  means,   is 
gradually  introduced  into  our  reasonings:  Similar  objects  are 
readily  taken  to  be  the  same:  And  the  conclusion  becomes  at 

last  very  wide  off  the  premises. — HUME,  DAVID. 

An  Inquiry  concerning  Human  Understanding, 
sect.  7,  part  1. 

1439.  One  part  of  these  disadvantages  in  moral  ideas  which 
has  made  them  be  thought  not  capable  of  demonstration,  may 
in  a  good  measure  be  remedied  by  definitions,  setting  down  that 
collection  of  simple  ideas,  which  every  term  shall  stand  for;  and 
then  using  the  terms  steadily  and  constantly  for  that  precise 
collection.    And  what  methods  algebra,  or  something  of  that 
kind,  may  hereafter  suggest,  to  remove  the  other  difficulties, 
it  is  not  easy  to  foretell.    Confident,  I  am,  that  if  men  would 
in  the  same  method,  and  with  the  same  indifferency,  search 
after  moral  as  they  do  mathematical  truths,  they  would  find 
them  have  a  stronger  connexion  one  with  another,  and  a  more 
necessary  consequence  from  our  clear  and  distinct  ideas,  and  to 
come  nearer  perfect  demonstration  than  is  commonly  imagined. 

LOCKE,  JOHN. 

An  Essay  concerning  Human  Understanding, 
Bk.  4,  chap.  3,  sect.  20. 


222  MEMORABILIA   MATHEMATICA 

1440.  That  which  in  this  respect  has  given  the  advantage  to 
the  ideas  of  quantity,  and  made  them  thought  more  capable  of 
certainty  and  demonstration  [than  moral  ideas],  is, 

First,  That  they  can  be  set  down  and  represented  by  sensible 
marks,  which  have  a  greater  and  nearer  correspondence  with 
them  than  any  words  or  sounds  whatsoever.  Diagrams  drawn 
on  paper  are  copies  of  the  ideas  in  the  mind,  and  not  liable  to  the 
uncertainty  that  words  carry  in  their  signification.  An  angle, 
circle,  or  square,  drawn  in  lines,  lies  open  to  the  view,  and 
cannot  be  mistaken:  it  remains  unchangeable,  and  may  at 
leisure  be  considered  and  examined,  and  the  demonstration  be 
revised,  and  all  the  parts  of  it  may  be  gone  over  more  than  once, 
without  any  danger  of  the  least  change  in  the  ideas.  This 
cannot  be  done  in  moral  ideas :  we  have  no  sensible  marks  that 
resemble  them,  whereby  we  can  set  them  down;  we  have  nothing 
but  words  to  express  them  by;  which,  though  when  written  they 
remain  the  same,  yet  the  ideas  they  stand  for  may  change  in  the 
same  man;  and  it  is  seldom  that  they  are  not  different  in  differ- 
ent persons. 

Secondly,  Another  thing  that  makes  the  greater  difficulty  in 
ethics  is,  That  moral  ideas  are  commonly  more  complex  than 
those  of  the  figures  ordinarily  considered  in  mathematics. 
From  whence  these  two  inconveniences  follow: — First,  that 
their  names  are  of  more  uncertain  signification,  the  precise 
collection  of  simple  ideas  they  stand  for  not  being  so  easily 
agreed  on;  and  so  the  sign  that  is  used  for  them  in  communica- 
tion always,  and  in  thinking  often,  does  not  steadily  carry  with 
it  the  same  idea.  Upon  which  the  same  disorder,  confusion,  and 
error  follow,  as  would  if  a  man,  going  to  demonstrate  something 
of  an  heptagon,  should,  in  the  diagram  he  took  to  do  it,  leave 
out  one  of  the  angles,  or  by  oversight  make  the  figure  with  an 
angle  more  than  the  name  ordinarily  imported,  or  he  intended 
it  should  when  at  first  he  thought  of  his  demonstration.  This 
often  happens,  and  is  hardly  avoidable  hi  very  complex  moral 
ideas,  where  the  same  name  being  retained,  an  angle,  i.  e.  one 
simple  idea  is  left  out,  or  put  in  the  complex  one  (still  called  by 
the  same  name)  more  at  one  tune  than  another.  Secondly, 
From  the  complexedness  of  these  moral  ideas  there  follows  an- 
other inconvenience,  viz.,  that  the  mind  cannot  easily  retain 


MATHEMATICS   AND    PHILOSOPHY  223 

those  precise  combinations  so  exactly  and  perfectly  as  is  neces- 
sary in  the  examination  of  the  habitudes  and  correspondences, 
agreements  or  disagreements,  of  several  of  them  one  with 
another;  especially  where  it  is  to  be  judged  of  by  long  deductions 
and  the  intervention  of  several  other  complex  ideas  to  show  the 
agreement  or  disagreement  of  two  remote  ones. — LOCKE,  JOHN. 

An  Essay  concerning  Human  Understanding, 

Bk.  4,  chap.  3,  sect.  19. 

1441.  It  has  been  generally  taken  for  granted,  that  mathe- 
matics alone  are  capable  of  demonstrative  certainty:  but  to 
have  such  an  agreement  or  disagreement  as  may  be  intuitively 
perceived,  being,  as  I  imagine,  not  the  privileges  of  the  ideas  of 
number,  extension,  and  figure  alone,  it  may  possibly  be  the  want 
of  due  method  and  application  in  us,  and  not  of  sufficient 
evidence  in  things,  that  demonstration  has  been  thought  to  have 
so  little  to  do  hi  other  parts  of  knowledge,  and  been  scarce  so 
much  as  aimed  at  by  any  but  mathematicians.    For  whatever 
ideas  we  have  wherein  the  mind  can  perceive  the  immediate 
agreement  or  disagreement  that  is  between  them,  there  the 
mind  is  capable  of  intuitive  knowledge,  and  where  it  can  per- 
ceive the  agreement  or  disagreement  of  any  two  ideas,  by  an 
intuitive  perception  of  the  agreement  or  disagreement  they  have 
with  any  intermediate  ideas,  there  the  mind  is  capable  of 
demonstration:  which  is  not  limited  to  the  idea  of  extension, 
figure,  number,  and  their  modes. — LOCKE,  JOHN. 

An  Essay  concerning  Human  Understanding, 
Bk.  4,  chap.  2,  sect.  9. 

1442.  Now  I  shall  remark  again  what  I  have  already  touched 
upon  more  than  once,  that  it  is  a  common  opinion  that  only 
mathematical  sciences  are  capable  of  a  demonstrative  cer- 
tainty; but  as  the  agreement  and  disagreement  which  may  be 
known  intuitively  is  not  a  privilege  belonging  only  to  the  ideas  of 
numbers  and  figures,  it  is  perhaps  for  want  of  application  on 
our  part  that  mathematics  alone  have  attained  to  demonstra- 
tions.— LEIBNITZ. 

New  Essay  concerning  Human  Understanding, 
Bk.  4,  chap.  2,  sect.  9  [Langley], 


CHAPTER  XV 

MATHEMATICS  AND  SCIENCE 

1501.  How  comes  it  about  that   the   knowledge  of  other 
sciences,  which  depend  upon  this  [mathematics],  is  painfully 
sought,  and  that  no  one  puts  himself  to  the  trouble  of  studying 
this  science  itself?    I  should  certainly  be  surprised,  if  I  did  not 
know  that  everybody  regarded  it  as  being  very  easy,  and  if  I 
had  not  long  ago  observed  that  the  human  mind,  neglecting 
what  it  believes  to  be  easy,  is  always  in  haste  to  run  after  what 
is  novel  and  advanced. — DESCARTES. 

Rules  for  the  Direction  of  the  Mind;  Philosophy 
of  Descartes  [Torrey],  (New  York,  1892),  p.  72. 

1502.  All  quantitative  determinations  are  in  the  hands  of 
mathematics,  and  it  at  once  follows  from  this  that  all  specula- 
tion which  is  heedless  of  mathematics,  which  does  not  enter 
into  partnership  with  it,  which  does  not  seek  its  aid  in  dis- 
tinguishing between  the  manifold  modifications  that  must  of 
necessity  arise  by  a  change  of  quantitative  determinations,  is 
either  an  empty  play  of  thoughts,  or  at  most  a  fruitless  effort. 
In  the  field  of  speculation  many  things  grow  which  do  not  start 
from  mathematics  nor  give  it  any  care,  and  I  am  far  from  assert- 
ing that  all  that  thus  grow  are  useless  weeds,  among  them  may  be 
many  noble  plants,  but  without  mathematics  none  will  develop 
to  complete  maturity. — HERBABT,  J.  F. 

Werke  (Kehrbach),  (Langensalza,  1890),  Bd.  5, 
p.  106. 

1503.  There  are  few  things  which  we  know,  which  are  not 
capable  of  being  reduc'd  to  a  Mathematical  Reasoning,  and 
when  they  cannot,  it's  a  sign  our  knowledge  of  them  is  very 
small  and  confus'd;  and  where  a  mathematical  reasoning  can 
be  had,  it's  as  great  folly  to  make  use  of  any  other,  as  to  grope 
for  a  thing  hi  the  dark,  when  you  have  a  candle  standing  by  you. 

ARBUTHNOT. 

Quoted  in  Todhunder's  History  of  the  Theory  of 
Probability  (Cambridge  and  London,  1865), 
p.  51. 

224 


MATHEMATICS   AND   SCIENCE  225 

1504.  Mathematical  Analysis  is  ...  the  true  rational  basis 
of  the  whole  system  of  our  positive  knowledge. — COMTE,  A. 

Positive  Philosophy  [Martineau],  Bk.  1,  chap.  1. 

1505.  It  is  only  through  Mathematics  that  we  can  thoroughly 
understand  what  true  science  is.    Here  alone  we  can  find  in  the 
highest  degree  simplicity  and  severity  of  scientific  law,  and  such 
abstraction  as  the  human  mind  can  attain.     Any  scientific 
education  setting  forth  from  any  other  point,  is  faulty  in  its 
basis. — COMTE,  A. 

Positive  Philosophy  [Martineau],  Bk.  1,  chap.  1. 

1506.  In  the  present  state  of  our  knowledge  we  must  regard 
Mathematics  less  as  a  constituent  part  of  natural  philosophy 
than  as  having  been,  since  the  time  of  Descartes  and  Newton, 
the  true  basis  of  the  whole  of  natural  philosophy;  though  it  is, 
exactly  speaking,  both  the  one  and  the  other.     To  us  it  is  of 
less  use  for  the  knowledge  of  which  it  consists,  substantial  and 
valuable  as  that  knowledge  is,  than  as  being  the  most  powerful 
instrument  that  the  human  mind  can  employ  in  the  investiga- 
tion of  the  laws  of  natural  phenomena. — COMTE,  A. 

Positive  Philosophy  [Martineau],  Introduction, 
chap.  2. 

1507.  The  concept  of  mathematics  is  the  concept  of  science 
in  general. — NOVALIS. 

Schriften  (Berlin,  1901),   Teil  2,  p.  222. 

1508.  I  contend,  that  each  natural  science  is  real  science  only 
in  so  far  as  it  is  mathematical.  ...  It  may  be  that  a  pure 
philosophy  of  nature  in  general  (that  is,  a  philosophy  which 
concerns  itself  only  with  the  general  concepts  of  nature)  is 
possible  without  mathematics,  but  a  pure  science  of  nature  deal- 
ing with  definite  objects  (physics  or  psychology),  is  possible 
only  by  means  of  mathematics,  and  since  each  natural  science 
contains  only  as  much  real  science  as  it  contains  a  priori  knowl- 
edge, each  natural  science  becomes  real  science  only  to  the  ex- 
tent that  it  permits  the  application  of  mathematics.  — KANT,  E. 

Metaphysische  Anfangsgriinde  der  Naturwis- 
senschaft,  Vorrede. 


226  MEMORABILIA   MATHEMATICA 

1509.  The   theory   most  prevalent  among  teachers  is  that 
mathematics  affords  the  best  training  for  the  reasoning  pow- 
ers; .  .  .  The  modem,  and  to  my  mind  true,  theory  is  that 
mathematics  is  the  abstract  form  of  the  natural  sciences;  and 
that  it  is  valuable  as  a  training  of  the  reasoning  powers,  not 
because  it  is  abstract,  but  because  it  is  a  representation  of 
actual  things. — SAFFORD,  T.  H. 

Mathematical  Teaching  etc.  (Boston,  1886),  p.  9. 

1510.  It  seems  to  me  that  no  one  science  can  so  well  serve  to 
co-ordinate  and,  as  it  were,  bind  together  all  of  the  sciences  as 
the  queen  of  them  all,  mathematics. — DAVIS,  E.  W. 

Proceedings  Nebraska  Academy  of  Sciences  for 
1896  (Lincoln,  1897),  p.  282. 

1511.  And  as  for  Mixed  Mathematics,  I  may  only  make  this 
prediction,  that  there  cannot  fail  to  be  more  kinds  of  them,  as 
nature  grows  further  disclosed. — BACON,  FRANCIS. 

Advancement  of  Learning,  Bk.  2;  De  Augmen- 
ts, Bk.  3. 

1512.  Besides  the  exercise  in  keen  comprehension  and  the 
certain  discovery  of  truth,  mathematics  has  another  formative 
function,  that  of  equipping  the  mind  for  the  survey  of  a  scientific 
system. — GRASSMANN,  H. 

Stiicke  aus  dem  Lehrbuche  der  Arithmetik; 
Werke  (Leipzig,  1904),  Bd.  2,  p.  298. 

1513.  Mathematicks  may  help  the  naturalists,  both  to  frame 
hypotheses,  and  to  judge  of  those  that  are  proposed  to  them, 
especially  such  as  relate  to  mathematical  subjects  hi  conjunction 
with  others. — BOYLE,  ROBERT. 

Works  (London,  1772),  Vol.  S,  p.  J&9. 

1514.  The  more  progress  physical  sciences  make,  the  more 
they  tend  to  enter  the  domain  of  mathematics,  which  is  a  kind  of 
centre  to  which  they  all  converge.    We  may  even  judge  of  the 
degree  of  perfection  to  which  a  science  has  arrived  by  the 
facility  with  which  it  may  be  submitted  to  calculation. 

QUETELET. 

Quoted  in  E.  MaiUy's  Eulogy  on  Quetelet; 
Smithsonian  Report,  1874,  P- 178. 


MATHEMATICS   AND    SCIENCE  227 

1515.  The  mathematical  formula  is  the  point  through  which 
all  the  light  gained  by  science  passes  hi  order  to  be  of  use  to 
practice;  it  is  also  the  point  in  which  all  knowledge  gained  by 
practice,  experiment,  and  observation  must  be  concentrated 
before  it  can  be  scientifically  grasped.    The  more  distant  and 
marked  the  point,  the  more  concentrated  will  be  the  light  com- 
ing from  it,  the  more  unmistakable  the  insight  conveyed.    All 
scientific   thought,   from  the   simple   gravitation   formula  of 
Newton,  through  the  more  complicated  formulae  of  physics 
and  chemistry,  the  vaguer  so  called  laws  of  organic  and  animated 
nature,  down  to  the  uncertain  statements  of  psychology  and  the 
data  of  our  social  and  historical  knowledge,  alike  partakes  of 
this  characteristic,  that  it  is  an  attempt  to  gather  up  the  scat- 
tered rays  of  light,  the  different  parts  of  knowledge,  in  a  focus, 
from  whence  it  can  be  again  spread  out  and  analyzed,  according 
to  the  abstract  processes  of  the  thinking  mind.    But  only  when 
this  can  be  done  with  a  mathematical  precision  and  accuracy  is 
the  image  sharp  and  well-defined,  and  the  deductions  clear  and 
unmistakable.    As  we  descend  from  the  mechanical,  through 
the  physical,  chemical,  and  biological,  to  the  mental,  moral,  and 
social  sciences,  the  process  of  focalization  becomes  less  and  less 
perfect, — the  sharp  point,  the  focus,  is  replaced  by  a  larger  or 
smaller  circle,  the  contours  of  the  image  become  less  and  less 
distinct,  and  with  the  possible  light  which  we  gain  there  is 
mingled  much  darkness,  the  sources  of  many  mistakes  and 
errors.     But  the  tendency  of  all  scientific  thought  is  toward 
clearer  and  clearer  definition;  it  lies  in  the  direction  of  a  more 
and  more  extended  use  of  mathematical  measurements,   of 
mathematical  formulae. — MERZ,  J.  T. 

History  of  European  Thought  in  the  19th 
Century  (Edinburgh  and  London,  1904), 
Vol.  1,  p.  333. 

1516.  From  the  very  outset  of  his  investigations  the  physicist 
has  to  rely  constantly  on  the  aid  of  the  mathematician,  for 
even  in  the  simplest  cases,  the  direct  results  of  his  measuring 
operations  are  entirely  without  meaning  until  they  have  been 
submitted  to  more  or  less  of  mathematical  discussion.     And 
when  in  this  way  some  interpretation  of  the  experimental 
results  has  been  arrived  at,  and  it  has  been  proved  that  two  or 


228  MEMORABILIA   MATHEMATICA 

more  physical  quantities  stand  in  a  definite  relation  to  each 
other,  the  mathematician  is  very  often  able  to  infer,  from  the 
existence  of  this  relation,  that  the  quantities  in  question  also 
fulfill  some  other  relation,  that  was  previously  unsuspected. 
Thus  when  Coulomb,  combining  the  functions  of  experimentalist 
and  mathematician,  had  discovered  the  law  of  the  force  exerted 
between,  two  particles  of  electricity,  it  became  a  purely  mathe- 
matical problem,  not  requiring  any  further  experiment,  to 
ascertain  how  electricity  is  distributed  upon  a  charged  conductor 
and  this  problem  has  been  solved  by  mathematicians  hi  several 
cases. — FOSTER,  G.  C. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science,  Section  A  (1877); 
Nature,  Vol.  16,  p.  812-813. 

1517.  Without  consummate  mathematical  skill,  on  the  part 
of  some  investigators  at  any  rate,  all  the  higher  physical  prob- 
lems would  be  sealed  to  us;  and  without  competent  skill  on 
the  part  of  the  ordinary  student  no  idea  can  be  formed  of  the 
nature  and  cogency  of  the  evidence  on  which  the  solutions  rest. 
Mathematics  are  not  merely  a  gate  through  which  we  may 
approach  if  we  please,  but  they  are  the  only  mode  of  approach 
to  large  and  important  districts  of  thought. — VENN,  JOHN. 

Symbolic  Logic  (London  and  New  York,  1894), 
Introduction,  p.  xix. 

1518.  Much  of  the  skill  of  the  true  mathematical  physicist 
and  of  the  mathematical  astronomer  consists  hi  the  power  of 
adapting  methods  and  results  carried  out  on  an  exact  mathe- 
matical basis  to  obtain  approximations  sufficient  for  the  pur- 
poses of  physical  measurements.    It  might  perhaps  be  thought 
that  a  scheme  of  Mathematics  on  a  frankly  approximative 
basis  would  be  sufficient  for  all  the  practical  purposes  of  applica- 
tion in  Physics,  Engineering  Science,  and  Astronomy,  and  no 
doubt  it  would  be  possible  to  develop,  to  some  extent  at  least,  a 
species  of  Mathematics  on  these  lines.    Such  a  system  would, 
however,  involve  an  intolerable  awkwardness  and  prolixity  in 
the  statements  of  results,  especially  in  view  of  the  fact  that  the 
degree  of  approximation  necessary  for  various  purposes  is  very 
different,  and  thus  that  unassigned  grades  of  approximation 


MATHEMATICS   AND    SCIENCE  229 

would  have  to  be  provided  for.  Moreover,  the  mathematician 
working  on  these  lines  would  be  cut  off  from  the  chief  sources  of 
inspiration,  the  ideals  of  exactitude  and  logical  rigour,  as  well  as 
from  one  of  his  most  indispensable  guides  to  discovery,  symme- 
try, and  permanence  of  mathematical  form.  The  history  of  the 
actual  movements  of  mathematical  thought  through  the  cen- 
turies shows  that  these  ideals  are  the  very  life-blood  of  the 
science,  and  warrants  the  conclusion  that  a  constant  striving 
toward  their  attainment  is  an  absolutely  essential  condition  of 
vigorous  growth.  These  ideals  have  their  roots  in  irresistible 
impulses  and  deep-seated  needs  of  the  human  mind,  manifested 
in  its  efforts  to  introduce  intelligibility  in  certain  great  domains 
of  the  world  of  thought. — HOBSON,  E.  W. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science,  Section  A  (1910}; 
Nature,  Vol.  84,  pp.  285-286. 

1519.  The  immense  part  which  those  laws  [laws  of  number 
and  extension]  take  in  giving  a  deductive  character  to  the  other 
departments  of  physical  science,  is  well  known;  and  is  not 
surprising,  when  we  consider  that  all  causes  operate  according 
to  mathematical  laws.    The  effect  is  always  dependent  upon,  or 
in  mathematical  language,  is  a  function  of,  the  quantity  of  the 
agent;  and  generally  of  its  position  also.    We  cannot,  therefore, 
reason  respecting  causation,  without  introducing  considerations 
of  quantity  and  extension  at  every  step;  and  if  the  nature  of  the 
phenomena  admits  of  our  obtaining  numerical  data  of  sufficient 
accuracy,  the  laws  of  quantity  become  the  grand  instruments  for 
calculating  forward  to  an  effect,  or  backward  to  a  cause. 

MILL,  J.  S. 
System  of  Logic,  Bk.  3,  chap.  24,  sect.  9. 

1520.  The  ordinary  mathematical  treatment  of  any  applied 
science  substitutes  exact  axioms  for  the  approximate  results  of 
experience,  and  deduces  from  these  axioms  the  rigid  mathe- 
matical conclusions.    In  applying  this  method  it  must  not  be 
forgotten  that  the  mathematical  developments  transcending  the 
limits  of  exactness  of  the  science  are  of  no  practical  value.    It 
follows  that  a  large  portion  of  abstract  mathematics  remains 
without  finding  any  practical  application,  the  amount  of  mathe- 


230  MEMORABILIA   MATHEMATICA 

matics  that  can  be  usefully  employed  in  any  science  being  in 
proportion  to  the  degree  of  accuracy  attained  in  the  science. 
Thus,  while  the  astronomer  can  put  to  use  a  wide  range  of 
mathematical  theory,  the  chemist  is  only  just  beginning  to 
apply  the  first  derivative,  i.  e.  the  rate  of  change  at  which 
certain  processes  are  going  on;  for  second  derivatives  he  does 
not  seem  to  have  found  any  use  as  yet. — KLEIN,  F. 

Lectures  on  Mathematics  (New  York,  1911), 

p.  47. 

1521.  The  bond  of  union  among  the  physical  sciences  is  the 
mathematical   spirit   and   the   mathematical   method   which 
pervades  them.  ...  Our  knowledge  of  nature,  as  it  advances, 
continuously  resolves  differences  of  quality  into  differences  of 
quantity.     All  exact  reasoning — indeed  all  reasoning — about 
quantity  is  mathematical  reasoning;  and  thus  as  our  knowledge 
increases,  that  portion  of  it  which  becomes  mathematical  in- 
creases at  a  still  more  rapid  rate. — SMITH,  H.  J.  S. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science,  Section  A  (1873}; 
Nature,  Vol.  8,  p.  449. 

1522.  Another  way  of  convincing  ourselves  how  largely  this 
process  [of  assimilation  of  mathematics  by  physics]  has  gone  on 
would  be  to  try  to  conceive  the  effect  of  some  intellectual 
catastrophe,    supposing   such   a   thing  possible,    whereby   all 
knowledge  of  mathematics  should  be  swept  away  from  men's 
minds.    Would  it  not  be  that  the  departure  of  mathematics 
would   be   the   destruction   of   physics?     Objective   physical 
phenomena  would,  indeed,  remain  as  they  are  now,  but  physical 
science  would  cease  to  exist.    We  should  no  doubt  see  the  same 
colours  on  looking  into  a  spectroscope  or  polariscope,  vibrating 
strings  would  produce  the  same  sounds,  electrical  machines 
would  give  sparks,  and  galvanometer  needles  would  be  de- 
flected; but  all  these  things  would  have  lost  their  meaning;  they 
would  be  but  as  the  dry  bones — the  disjecta  membra — of  what 
is  now  a  living  and  growing  science.    To  follow  this  conception 
further,  and  to  try  to  image  to  ourselves  in  some  detail  what 
would  be  the  kind  of  knowledge  of  physics  which  would  remain 
possible,  supposing  all  mathematical  ideas  to  be  blotted  out, 


MATHEMATICS   AND    SCIENCE  231 

would  be  extremely  interesting,  but  it  would  lead  us  directly 
into  a  dim  and  entangled  region  where  the  subjective  seems  to  be 
always  passing  itself  off  for  the  objective,  and  where  I  at  least 
could  not  attempt  to  lead  the  way,  gladly  as  I  would  follow  any 
one  who  could  show  where  a  firm  footing  is  to  be  found.  But 
without  venturing  to  do  more  than  to  look  from  a  safe  distance 
over  this  puzzling  ground,  we  may  see  clearly  enough  that 
mathematics  is  the  connective  tissue  of  physics,  binding  what 
would  else  be  merely  a  list  of  detached  observations  into  an 
organized  body  of  science. — FOSTER,  G.  C. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science,  Section  A  (1877); 
Nature,  Vol.  16,  p.  SIS. 

1523.  In  Plato's  time  mathematics  was  purely  a  play  of  the 
free  intellect;  the  mathematic-mystical  reveries  of  a  Pythagoras 
foreshadowed  a  far-reaching  significance,  but  such  a  significance 
(except  in  the  case  of  music)  was  as  yet  entirely  a  matter  of 
fancy;  yet  even  in  that  time  mathematics  was  the  prerequisite 
to  all  other  studies!    But  today,  when  mathematics  furnishes 
the  only  language  by  means  of  which  we  may  formulate  the 
most  comprehensive  laws  of  nature,  laws  which  the  ancients 
scarcely  dreamed  of,  when  moreover  mathematics  is  the  only 
means  by  which  these  laws  may  be  understood, — how  few  learn 
today  anything  of  the  real  essence  of  our  mathematics!  .  .  . 
In  the  schools  of  today  mathematics  serves  only  as  a  disciplinary 
study,  a  mental  gymnastic;  that  it  includes  the  highest  ideal 
value  for  the  comprehension  of  the  universe,  one  dares  scarcely 
to  think  of  hi  view  of  our  present  day  instruction. 

LINDEMAN,  F. 

Lehren  und  Lernen  in  der  Mathematik  (Miin- 
chen,  1904),  P-  *4- 

1524.  All  applications  of  mathematics  consist  in  extending  the 
empirical  knowledge  which  we  possess  of  a  limited  number  or 
region  of  accessible  phenomena  into  the  region  of  the  unknown 
and  inaccessible;  and  much  of  the  progress  of  pure  analysis 
consists  in  inventing  definite  conceptions,  marked  by  symbols, 
of  complicated  operations;  in  ascertaining  their  properties  as 
independent  objects  of  research;  and  in  extending  their  meaning 


232  MEMORABILIA    MATHEMATICA 

beyond  the  limits  they  were  originally  invented  for, — thus 
opening  out  new  and  larger  regions  of  thought. — MERZ,  J.  T. 
History  of  European   Thought  in   the    19th 
Century  (Edinburgh  and  London,  1908),  Vol.  1, 
p.  698. 

1526.  All  the  effects  of  nature  are  only  mathematical  results 
of  a  small  number  of  immutable  laws. — LAPLACE. 

A  Philosophical  Essay  on  Probabilities  [Trus- 
cott  and  Emory]  (New  York,  1902),  p.  177; 
Oeuvres,  t.  7,  p.  189. 

1526.  What  logarithms  are  to  mathematics  that  mathe- 
matics are  to  the  other  sciences. — NOVALIS. 

Schriften  (Berlin,  1901),  Teil  2,  p.  222. 

1527.  Any  intelligent  man  may  now,  by  resolutely  applying 
himself  for  a  few  years  to  mathematics,  learn  more  than  the 
great  Newton  knew  after  half  a  century  of  study  and  meditation. 

MACAULAT. 

Milton;  Critical  and  Miscellaneous  Essays 
(New  York,  1879),  Vol.  1,  p.  18. 

1528.  In  questions  of  science  the  authority  of  a  thousand  is 
not  worth  the  humble  reasoning  of  a  single  individual. 

GALILEO. 

Quoted  in  Arago's  Eulogy  on  Laplace;  Smith- 
sonian Report,  1874,  P- 164. 

1529.  Behind  the  artisan  is  the  chemist,  behind  the  chemist  a 
physicist,  behind  the  physicist  a  mathematician. — WHITE,  W.  F. 

Scrap-book  of  Elementary  Mathematics  (Chi- 
cago, 1908),  p.  217. 

1530.  The  advance  in  our  knowledge  of  physics  is  largely  due 
to  the  application  to  it  of  mathematics,  and  every  year  it  be- 
comes more  difficult  for  an  experimenter  to  make  any  mark  in 
the  subject  unless  he  is  also  a  mathematician. — BALL,  W.  W.  R. 

History  of  Mathematics  (London,  1901),  p.  508. 

1531.  In  very  many  cases  the  most  obvious  and  direct 
experimental  method  of  investigating  a  given  problem  is  ex- 
tremely difficult,  or  for  some  reason  or  other  untrustworthy. 


MATHEMATICS   AND    SCIENCE  233 

In  such  cases  the  mathematician  can  often  point  out  some  other 
problem  more  accessible  to  experimental  treatment,  the  solu- 
tion of  which  involves  the  solution  of  the  former  one.  For 
example,  if  we  try  to  deduce  from  direct  experiments  the  law 
according  to  which  one  pole  of  a  magnet  attracts  or  repels  a 
pole  of  another  magnet,  the  observed  action  is  so  much  com- 
plicated with  the  effects  of  the  mutual  induction  of  the  magnets 
and  of  the  forces  due  to  the  second  pole  of  each  magnet,  that 
it  is  next  to  impossible  to  obtain  results  of  any  great  accuracy. 
Gauss,  however,  showed  how  the  law  which  applied  in  the 
case  mentioned  can  be  deduced  from  the  deflections  undergone 
by  a  small  suspended  magnetic  needle  when  it  is  acted  upon  by  a 
small  fixed  magnet  placed  successively  in  two  determinate 
positions  relatively  to  the  needle;  and  being  an  experimentalist 
as  well  as  a  mathematician,  he  showed  likewise  how  these 
deflections  can  be  measured  very  easily  and  with  great  precision. 

FOSTER,  G.  C. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science,  Section  A  (1877); 
Nature,  Vol.  16,  p.  818. 

1532.    Give  me  to  learn  each  secret  cause; 

Let  Number's,  Figure's,  Motion's  laws 
Reveal'd  before  me  stand; 
These  to  great  Nature's  scenes  apply, 
And  round  the  globe,  and  through  the  sky, 

Disclose  her  working  hand. — AKENSIDE,  M. 

Hymn  to  Science. 

1633.  Now  there  are  several  scores,  upon  which  skill  in 
mathematicks  may  be  useful  to  the  experimental  philosopher. 
For  there  are  some  general  advantages,  which  mathematicks 
may  bring  to  the  minds  of  men,  to  whatever  study  they  apply 
themselves,  and  consequently  to  the  student  of  natural  philos- 
ophy; namely,  that  these  disciplines  are  wont  to  make  men 
accurate,  and  very  attentive  to  the  employment  that  they  are 
about,  keeping  their  thoughts  from  wandering,  and  muring 
them  to  patience  in  going  through  with  tedious  and  intricate 
demonstrations;  besides,  that  they  much  improve  reason,  by 
accustoming  the  mind  to  deduce  successive  consequences, 


234  MEMORABILIA   MATHEMATICA 

and  judge  of  them  without  easily  acquiescing  in  anything  but 
demonstration. — BOYLE,  ROBERT. 

Works  (London,  1772\  Vol.  3,  p.  426. 

1534.  It  is  not  easy  to  anatomize  the  constitution  and  the 
operations  of  a  mind  [like  Newton's]  which  makes  such  an 
advance  in  knowledge.    Yet  we  may  observe  that  there  must 
exist  hi  it,  in  an  eminent  degree,  the  elements  which  compose  the 
mathematical  talent.    It  must  possess  distinctness  of  intuition, 
tenacity  and  facility  in  tracing  logical  connection,  fertility  of 
invention,  and  a  strong  tendency  to  generalization. 

WHEWELL,  W. 

History  of  the  Inductive  Sciences  (New  York, 
1894),  Vol.  1,  p.  416. 

1535.  The  domain  of  physics  is  no  proper  field  for  mathe- 
matical pastimes.     The  best  security  would  be  in  giving  a 
geometrical  training  to  physicists,  who  need  not  then  have 
recourse  to  mathematicians,   whose  tendency  is  to   despise 
experimental  science.    By  this  method  will  that  union  between 
the  abstract  and  the  concrete  be  effected  which  will  perfect  the 
uses  of  mathematical,  while  extending  the  positive  value  of 
physical  science.    Meantime,  the  uses  of  analysis  hi  physics  is 
clear  enough.    Without  it  we  should  have  no  precision,  and  no 
co-ordination;  and  what  account  could  we  give  of  our  study  of 
heat,  weight,  light,  etc.?     We  should  have  merely  series  of 
unconnected  facts,  in  which  we  could  foresee  nothing  but  by 
constant  recourse  to  experiment;  whereas,  they  now  have  a 
character  of  rationality  which  fits  them  for  purposes  of  previ- 
sion.— COMTE,  A. 

Positive  Philosophy  [Martineau]  Bk.  8,  chap.  1 . 

1536.  It  must  ever  be  remembered  that  the  true  positive 
spirit  first  came  forth  from  the  pure  sources  of  mathematical 
science;  and  it  is  only  the  mind  that  has  imbibed  it  there,  and 
which  has  been  face  to  face  with  the  lucid  truths  of  geometry 
and  mechanics,  that  can  bring  into  full  action  its  natural 
positivity,  and  apply  it  hi  bringing  the  most  complex  studies  into 
the  reality  of  demonstration.    No  other  discipline  can  fitly  pre- 
pare the  intellectual  organ. — COMTE,  A. 

Positive  Philosophy,  [Martineau]  Bk.  8,  chap.  1 . 


MATHEMATICS   AND    SCIENCE  235 

1537.  During  the  last  two  centuries  and  a  half,  physical 
knowledge  has  been  gradually  made  to  rest  upon  a  basis  which 
it  had  not  before.    It  has  become  mathematical.    The  question 
now  is,  not  whether  this  or  that  hypothesis  is  better  or  worse  to 
the  pure  thought,  but  whether  it  accords  with  observed  phenom- 
ena hi  those  consequences  which  can  be  shown  necessarily  to 
follow  from  it,  if  it  be  true.    Even  in  those  sciences  which  are 
not  yet  under  the  dominion  of  mathematics,  and  perhaps  never 
will  be,  a  working  copy  of  the  mathematical  process  has  been 
made.    This  is  not  known  to  the  followers  of  those  sciences  who 
are  not  themselves  mathematicians,  and  who  very  often  exalt 
their  horns  against  the  mathematics  in  consequence.     They 
might  as  well  be  squaring  the  circle,  for  any  sense  they  show 
in  this  particular. — DE  MORGAN,  A. 

A  Budget  of  Paradoxes  (London,  1872),  p.  2. 

1538.  Among  the  mere  talkers  so  far  as  mathematics  are 
concerned,  are  to  be  ranked  three  out  of  four  of  those  who  apply 
mathematics  to  physics,  who,  wanting  a  tool  only,  are  very 
impatient  of  everything  which  is  not  of  direct  aid  to  the  actual 
methods  which  are  in  their  hands. — DE  MORGAN,  A. 

Graves'  Life  of  Sir  William  Rowan  Hamilton 
(New  York,  1882-1889),  Vol.  8,  p.  348. 

1539.  Something  has  been  said  about  the  use  of  mathematics 
in  physical  science,  the  mathematics  being  regarded  as  a  weapon 
forged  by  others,  and  the  study  of  the  weapon  being  com- 
pletely set  aside.    I  can  only  say  that  there  is  danger  of  obtaining 
untrustworthy  results  hi  physical  science,  if  only  the  results  of 
mathematics  are  used;  for  the  person  so  using  the  weapon  can 
remain  unacquainted  with  the  conditions  under  which  it  can  be 
rightly  applied.  .  .  .  The  results  are  often  correct,  sometimes 
are  incorrect;  the  consequence  of  the  latter  class  of  cases  is  to 
throw  doubt  upon  all  the  applications  of  such  a  worker  until  a 
result  has  been  otherwise  tested.    Moreover,  such  a  practice  in 
the  use  of  mathematics  leads  a  worker  to  a  mere  repetition  in  the 
use  of  familiar  weapons;  he  is  unable  to  adapt  them  with  any 
confidence  when  some  new  set  of  conditions  arise  with  a  demand 
for  a  new  method:  for  want  of  adequate  instruction  hi  the 


236  MEMORABILIA   MATHEMATICA 

forging  of  the  weapon,  he  may  find  himself,  sooner  or  later  in  the 
progress  of  his  subject,  without  any  weapon  worth  having. 

FORSYTH,  A.  R. 

Perry's  Teaching  of  Mathematics  (London, 
1902},  p.  36. 

1540.  If  in  the  range  of  human  endeavor  after  sound  knowl- 
edge there  is  one  subject  that  needs  to  be  practical,  it  surely  is 
Medicine.    Yet  in  the  field  of  Medicine  it  has  been  found  that 
branches  such  as  biology  and  pathology  must  be  studied  for 
themselves  and  be  developed  by  themselves  with  the  single  aim 
of  increasing  knowledge;  and  it  is  then  that  they  can  be  best 
applied  to  the  conduct  of  living  processes.    So  also  hi  the  pur- 
suit of  mathematics,  the  path  of  practical  utility  is  too  narrow 
and  irregular,  not  always  leading  far.    The  witness  of  history 
shows  that,  hi  the  field  of  natural  philosophy,  mathematics  will 
furnish  the  more  effective  assistance  if,  hi  its  systematic  develop- 
ment, its  course  can  freely  pass  beyond  the  ever-shifting  domain 
of  use  and  application. — FORSYTH,  A.  R. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science,  Section  A;  Nature, 
Vol.  56  (1897),  p.  877. 

1541.  If  the   Greeks   had  not   cultivated   Conic   Sections, 
Kepler  could  not  have  superseded  Ptolemy;  if  the  Greeks  had 
cultivated  Dynamics,  Kepler  might  have  anticipated  Newton. 

WHEWELL,  W. 

History  of  the  Inductive  Science  (New  York, 
1894),  Vol.  1,  p.  311. 

1542.  If  we  may  use  the  great  names  of  Kepler  and  Newton  to 
signify  stages  in  the  progress  of  human  discovery,  it  is  not  too 
much  to  say  that  without  the  treatises  of  the  Greek  geometers  on 
the  conic  sections  there  could  have  been  no  Kepler,  without 
Kepler  no  Newton,  and  without  Newton  no  science  in  the 
modern  sense  of  the  term,  or  at  least  no  such  conception  of 
nature  as  now  lies  at  the  basis  of  all  our  science,  of  nature  as 
subject  hi  the  smallest  as  well  as  hi  its  greatest  phenomena,  to 
exact  quantitative  relations,  and  to  definite  numerical  laws. 

SMITH,  H.  J.  S. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science,  Section  A;  Nature, 
Vol.  8  (1873),  p.  450. 


MATHEMATICS   AND   SCIENCE  237 

1543.  The  silent  work  of  the  great  Regiomontanus  in  his 
chamber  at  Nuremberg  computed  the  Ephemerides  which  made 
possible  the  discovery  of  America  by  Columbus. — RUDIO,  F. 

Quoted  in  Max  Simon's  Geschichte  der  Mathe- 
matik  im  Altertum  (Berlin,  1909),  Einleitung, 
p.  xi. 

1544.  The  calculation  of  the  eclipses  of  Jupiter's  satellites, 
many  a  man  might  have  been  disposed,  originally,  to  regard  as  a 
most  unprofitable  study.    But  the  utility  of  it  to  navigation  (in 
the  determination  of  longitudes)  is  now  well  known. 

WHATELY,  R. 

Annotations  to  Bacon's  Essays  (Boston,  1783), 
p.  492. 

1545.  Who  could  have  imagined,  when  Galvani  observed  the 
twitching  of  the  frog  muscles  as  he  brought  various  metals  in 
contact  with  them,  that  eighty  years  later  Europe  would  be 
overspun  with  wires  which  transmit  messages  from  Madrid  to 
St.  Petersburg  with  the  rapidity  of  lightning,  by  means  of  the 
same  principle  whose  first  manifestations  this  anatomist  then 
observed  !  .  .  . 

He  who  seeks  for  immediate  practical  use  in  the  pursuit  of 
science,  may  be  reasonably  sure,  that  he  will  seek  in  vain. 
Complete  knowledge  and  complete  understanding  of  the  action 
of  forces  of  nature  and  of  the  mind,  is  the  only  thing  that 
science  can  aim  at.  The  individual  investigator  must  find  his 
reward  in  the  joy  of  new  discoveries,  as  new  victories  of  thought 
over  resisting  matter,  in  the  esthetic  beauty  which  a  well- 
ordered  domain  of  knowledge  affords,  where  all  parts  are  in- 
tellectually related,  where  one  thing  evolves  from  another,  and 
all  show  the  marks  of  the  mind's  supremacy;  he  must  find  his 
reward  in  the  consciousness  of  having  contributed  to  the  grow- 
ing capital  of  knowledge  on  which  depends  the  supremacy  of 
man  over  the  forces  hostile  to  the  spirit. — HELMHOLTZ,  H. 
Vortrage  und  Reden  (Braunschweig,  1884), 
Bd.  1,  p.  142. 

1546.  When  the  time  comes  that  knowledge  will  not  be 
sought  for  its  own  sake,  and  men  will  not  press  forward  simply 


238  MEMORABILIA   MATHEMATICA 

in  a  desire  of  achievement,  without  hope  of  gain,  to  extend  the 
limits  of  human  knowledge  and  information,  then,  indeed,  will 
the  race  enter  upon  its  decadence. — HUGHES,  C.  E. 

Quoted  in  D.  E.  Smith's  Teaching  of  Geometry 

(Boston,  1911),  p.  9. 

1647.  [In  the  Opus  Majus  of  Roger  Bacon]  there  is  a  chapter, 
in  which  it  is  proved  by  reason,  that  all  sciences  require  mathe- 
matics. And  the  arguments  which  are  used  to  establish  this 
doctrine,  show  a  most  just  appreciation  of  the  office  of  mathe- 
matics in  science.  They  are  such  as  follows:  That  other  sciences 
use  examples  taken  from  mathematics  as  the  most  evident: — 
That  mathematical  knowledge  is,  as  it  were,  innate  to  us,  on 
which  point  he  refers  to  the  well-known  dialogue  of  Plato,  as 
quoted  by  Cicero: — That  this  science,  being  the  easiest,  offers 
the  best  introduction  to  the  more  difficult: — That  in  mathe- 
matics, things  as  known  to  us  are  identical  with  things  as 
known  to  nature: — That  we  can  here  entirely  avoid  doubt  and 
error,  and  obtain  certainty  and  truth: — That  mathematics  is 
prior  to  other  sciences  in  nature,  because  it  takes  cognizance  of 
quantity,  which  is  apprehended  by  intuition  (intuitu  intellectus) . 
"  Moreover,"  he  adds,  "  there  have  been  found  famous  men, 
as  Robert,  bishop  of  Lincoln,  and  Brother  Adam  Marshman 
(de  Marisco),  and  many  others,  who  by  the  power  of  mathe- 
matics have  been  able  to  explain  the  causes  of  things;  as  may  be 
seen  hi  the  writings  of  these  men,  for  instance,  concerning  the 
Rainbow  and  Comets,  and  the  generation  of  heat,  and  climates, 
and  the  celestial  bodies." — WHEWELL,  W. 

History  of  the  Inductive  Sciences  (New  York, 
1894),  Vol.  1,  p.  519.  Bacon,  Roger:  Opus 
Majus,  Part  4,  Distinctia  Prima,  cap.  3. 

1548.  The  analysis  which  is  based  upon  the  conception  of 
function  discloses  to  the  astronomer  and  physicist  not  merely 
the  formulae  for  the  computation  of  whatever  desired  distances, 
times,  velocities,  physical  constants;  it  moreover  gives  him 
insight  into  the  laws  of  the  processes  of  motion,  teaches  him  to 
predict  future  occurrences  from  past  experiences  and  supplies 
him  with  means  to  a  scientific  knowledge  of  nature,  i.  e.  it 
enables  him  to  trace  back  whole  groups  of  various,  sometimes 


MATHEMATICS   AND    SCIENCE  239 

extremely  heterogeneous,  phenomena  to  a  minimum  of  simple 
fundamental  laws. — PRINGSHEIM,  A. 

Jahresbericht  der  Deutschen  Mathematiker 
Vereinigung,  Bd.  18,  p.  866. 

1549.  "  As  is  known,  scientific  physics  dates  its  existence  from 
the  discovery  of  the  differential  calculus.    Only  when  it  was 
learned  how  to  follow  continuously  the  course  of  natural  events, 
attempts,  to  construct  by  means  of  abstract  conceptions  the 
connection  between  phenomena,  met  with  success.     To  do 
this  two  things  are  necessary:  First,  simple  fundamental  con- 
cepts with  which  to  construct;  second,  some  method  by  which  to 
deduce,  from  the  simple  fundamental  laws  of  the  construction 
which  relate  to  instants  of  time  and  points  in  space,  laws  for 
finite  intervals  and  distances,  which  alone  are  accessible  to 
observation  (can  be  compared  with  experience)."    [Riemann.] 

The  first  of  the  two  problems  here  indicated  by  Riemann  con- 
sists in  setting  up  the  differential  equation,  based  upon  physical 
facts  and  hypotheses.  The  second  is  the  integration  of  this 
differential  equation  and  its  application  to  each  separate  con- 
crete case,  this  is  the  task  of  mathematics. — WEBER,  HEINRICH. 

Die  partiellen  Differentialgleichungen  der 
mathematischen  Physik  (Braunschweig,  1882), 
Bd.  1,  Vorrede. 

1550.  Mathematics  is  the  most  powerful  instrument  which  we 
possess  for  this  purpose  [to  trace  into  their  farthest  results  those 
general  laws  which  an  inductive  philosophy  has  supplied]: 
in  many  sciences  a  profound  knowledge  of  mathematics  is 
indispensable  for  a  successful  investigation.    In  the  most  delicate 
researches  into  the  theories  of  light,  heat,  and  sound  it  is  the  only 
instrument;  they  have  properties  which  no  other  language  can 
express;  and  their  argumentative  processes  are  beyond  the 
reach  of  other  symbols. — PRICE,  B. 

Treatise  on  Infinitesimal  Calculus  (Oxford, 
1858),  Vol.  8,  p.  5. 

1551.  Notwithstanding  the  eminent  difficulties  of  the  mathe- 
matical theory  of  sonorous  vibrations,  we  owe  to  it  such  progress 
as  has  yet  been   made  in  acoustics.     The  formation  of  the 


240  MEMORABILIA   MATHEMATICA 

differential  equations  proper  to  the  phenomena  is,  independent 
of  their  integration,  a  very  important  acquisition,  on  account  of 
the  approximations  which  mathematical  analysis  allows  between 
questions,  otherwise  heterogeneous,  which  lead  to  similar 
equations.  This  fundamental  property,  whose  value  we  have  so 
often  to  recognize,  applies  remarkably  in  the  present  case;  and 
especially  since  the  creation  of  mathematical  thermology,  whose 
principal  equations  are  strongly  analogous  to  those  of  vibratory 
motion. — This  means  of  investigation  is  all  the  more  valuable 
on  account  of  the  difficulties  in  the  way  of  direct  inquiry  into 
the  phenomena  of  sound.  We  may  decide  the  necessity  of  the 
atmospheric  medium  for  the  transmission  of  sonorous  vibra- 
tions; and  we  may  conceive  of  the  possibility  of  determining  by 
experiment  the  duration  of  the  propagation,  in  the  air,  and  then 
through  other  media;  but  the  general  laws  of  the  vibrations  of 
sonorous  bodies  escape  immediate  observation.  We  should 
know  almost  nothing  of  the  whole  case  if  the  mathematical 
theory  did  not  come  in  to  connect  the  different  phenomena  of 
sound,  enabling  us  to  substitute  for  direct  observation  an 
equivalent  examination  of  more  favorable  cases  subjected  to  the 
same  law.  For  instance,  when  the  analysis  of  the  problem  of 
vibrating  chords  has  shown  us  that,  other  things  being  equal, 
the  number  of  oscillations  is  hi  inverse  proportion  to  the  length 
of  the  chord,  we  see  that  the  most  rapid  vibrations  of  a  very 
short  chord  may  be  counted,  since  the  law  enables  us  to  direct 
our  attention  to  very  slow  vibrations.  The  same  substitution 
is  at  our  command  in  many  cases  in  which  it  is  less  direct. 

COMTE,  A. 

Positive    Philosophy    [Martineau],     Bk.    3, 

chap.  4- 

1562.  Problems  relative  to  the  uniform  propagation,  or 
to  the  varied  movements  of  heat  in  the  interior  of  solids,  are 
reduced  ...  to  problems  of  pure  analysis,  and  the  progress  of 
this  part  of  physics  will  depend  in  consequence  upon  the  advance 
which  may  be  made  in  the  art  of  analysis.  The  differential 
equations  .  .  .  contain  the  chief  results  of  the  theory;  they 
express,  in  the  most  general  and  concise  manner,  the  necessary 
relations  of  numerical  analysis  to  a  very  extensive  class  of 


MATHEMATICS  AND   SCIENCE  241 

phenomena;  and  they  connect  forever  with  mathematical 
science  one  of  the  most  important  branches  of  natural  philos- 
ophy.— FOURIER,  J. 

Theory  of  Heat  [Freeman],  (Cambridge,  1878), 

Chap.  8,  p.  1S1. 

1553.  The  effects  of  heat  are  subject  to  constant  laws  which 
cannot  be  discovered  without  the  aid  of  mathematical  analysis. 
The  object  of  the  theory  is  to  demonstrate  these  laws;  it  reduces 
all  physical  researches  on  the  propagation  of  heat,  to  problems 
of  the  integral  calculus,  whose  elements  are  given  by  experiment. 
No  subject  has  more  extensive  relations  with  the  progress  of 
industry  and  the  natural  sciences;  for  the  action  of  heat  is 
always  present,  it  influences  the  processes  of  the  arts,  and  oc- 
curs in  all  the  phenomena  of  the  universe. — FOURIER,  J. 

Theory  of  Heat  [Freeman],  (Cambridge,  1878), 
Chap.  1,  p.  12. 

1554.  Dealing  with  any  and  every  amount  of  static  electricity, 
the  mathematical  mind  has  balanced  and  adjusted  them  with 
wonderful  advantage,  and  has  foretold  results  which  the  ex- 
perimentalist can  do  no  more  than  verify.  ...  So  in  respect  of 
the  force  of  gravitation,  it  has  calculated  the  results  of  the 
power  in  such  a  wonderful  manner  as  to  trace  the  known  planets 
through  their  courses  and  perturbations,  and  in  so  doing  has 
discovered  a  planet  before  unknown. — FARADAY. 

Some  Thoughts  on  the  Conservation  of  Force. 

1555.  Certain  branches  of  natural  philosophy  (such  as  physi- 
cal astronomy  and  optics),  .  .  .  are,  in  a  great  measure,  inacces- 
sible to  those  who  have  not  received  a  regular  mathematical 
education  .  .  . — STEWART,  DUGALD. 

Philosophy  of  the  Human  Mind,  Part  S,  chap.  1 , 
sect.  S. 

1556.  So  intimate  is  the  union  between  mathematics  and 
physics  that  probably  by  far  the  larger  part  of  the  accessions  to 
our  mathematical  knowledge  have  been  obtained  by  the  efforts 
of  mathematicians  to  solve  the  problems  set  to  them  by  experi- 
ment, and  to  create  "for  each  successive  class  of  phenomena,  a 
new  calculus  or  a  new  geometry,  as  the  case  might  be,  which 


242  MEMORABILIA   MATHEMATICA 

might  prove  not  wholly  inadequate  to  the  subtlety  of  nature." 
Sometimes,  indeed,  the  mathematician  has  been  before  the 
physicist,  and  it  has  happened  that  when  some  great  and  new 
question  has  occurred  to  the  experimentalist  or  the  observer, 
he  has  found  in  the  armoury  of  the  mathematician  the  weapons 
which  he  has  needed  ready  made  to  his  hand.  But,  much 
oftener,  the  questions  proposed  by  the  physicist  have  trans- 
cended the  utmost  powers  of  the  mathematics  of  the  tune,  and  a 
fresh  mathematical  creation  has  been  needed  to  supply  the 
logical  instrument  requisite  to  interpret  the  new  enigma. 

SMITH,  H.  J.  S. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science,  Section  A;  Nature, 
Vol.  8  (1878},  p.  450. 

1557.  Of  all  the  great  subjects  which  belong  to  the  province  of 
his  section,  take  that  which  at  first  sight  is  the  least  within  the 
domain  of  mathematics — I  mean  meteorology.     Yet  the  part 
which  mathematics  plays  in  meteorology  increases  every  year, 
and  seems  destined  to  increase.    Not  only  is  the  theory  of  the 
simplest  instruments  essentially  mathematical,  but  the  discus- 
sions of  the  observations — upon  which,   be  it  remembered, 
depend  the  hopes  which  are  already  entertained  with  increasing 
confidence,  of  reducing  the  most  variable  and  complex  of  all 
known  phenomena  to  exact  laws — is  a  problem  which  not  only 
belongs  wholly  to  mathematics,  but  which  taxes  to  the  utmost 
the  resources  of  the  mathematics  which  we  now  possess. 

SMITH,  H.  J.  S. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science,  Section  A;  Nature, 
Vol.  8  (1873),  p.  449. 

1558.  You  know  that  if  you  make  a  dot  on  a  piece  of  paper, 
and  then  hold  a  piece  of  Iceland  spar  over  it,  you  will  see  not  one 
dot  but  two.     A  mineralogist,  by  measuring  the  angles  of  a 
crystal,  can  tell  you  whether  or  no  it  possesses  this  property 
without  looking  through  it.    He  requires  no  scientific  thought  to 
do  that.     But "  Sir  William  Roman  Hamilton  .  .  .  knowing 
these  facts  and  also  the  explanation  of  them  which  Fresnel  had 


MATHEMATICS   AND    SCIENCE  243 

given,  thought  about  the  subject,  and  he  predicted  that  by 
looking  through  certain  crystals  in  a  particular  direction  we 
should  see  not  two  dots  but  a  continuous  circle.  Mr.  Lloyd 
made  the  experiment,  and  saw  the  circle,  a  result  which  had 
never  been  even  suspected.  This  has  always  been  considered  one 
of  the  most  signal  instances  of  scientific  thought  in  the  domain 
of  physics. — CLIFFORD,  W.  K. 

Lectures  and  Essays  (New  York,  1901),  Vol.  1, 

p.  144. 

1559.  The  discovery  of  this  planet  [Neptune]  is  justly  reck- 
oned as  the  greatest  triumph  of  mathematical   astronomy. 
Uranus  failed  to  move  precisely  in  the  path  which  the  compu- 
ters predicted  for  it,  and  was  misguided  by  some  unknown 
influence  to  an  extent  which  a  keen  eye  might  almost  see  without 
telescopic  aid.  .  .  .  These  minute  discrepancies  constituted  the 
data  which  were  found  sufficient  for  calculating  the  position  of  a 
hitherto  unknown  planet,  and  bringing  it  to  light.    Leverrier 
wrote  to  Galle,  in  substance:  "Direct  your  telescope  to  a  point 
on  the  ecliptic  in  the  constellation  of  Aquarius,  in  longitude  326°, 
and  you  will  find  within  a  degree  of  that  place  a  new  planet,  looking 
like  a  star  of  about  the  ninth  magnitude,  and  having  a  perceptible 
disc."    The  planet  was  found  at  Berlin  on  the  night  of  Sept.  26, 
1846,  in  exact  accordance  with  this  prediction,  within  half  an 
hour  after  the  astronomers  began  looking  for  it,  and  only  about 
52'  distant  from  the  precise  point  that  Leverrier  had  indicated. 

•     YOUNG,  C.  A. 
General  Astronomy  (Boston,  1891),  Art.  653. 

1560.  I  am  convinced  that  the  future  progress  of  chemistry  as 
an  exact  science  depends  very  much  indeed  upon  the  alliance 
with  mathematics. — FRANKLAND,  A. 

American  Journal  of  Mathematics,   Vol.   1, 
p.  349. 

1561.  It  is  almost  impossible  to  follow  the  later  developments 
of  physical  or  general  chemistry  without  a  working  knowledge 
of  higher  mathematics. — MELLOR,  J.  W. 

Higher  Mathematics  (New  York,  1902},  Preface. 


244  MEMORABILIA   MATHEMATICA 

1562.  .  .  .  Mount  where  science  guides; 

Go  measure  earth,  weigh  air,  and  state  the  tides; 
Instruct  the  planets  in  what  orb  to  run, 
Correct  old  time,  and  regulate  the  sun. — THOMSON,  W. 
On  the  Figure  of  the  Earth,  Title  page. 

1563.  Admission  to  its  sanctuary  [referring  to  astronomy]  and 
to  the  privileges  and  feelings  of  a  votary,  is  only  to  be  gained  by 
one  means, — sound  and  sufficient  knowledge  of  mathematics,  the 
great  instrument  of  all  exact  inquiry,  without  which  no  man  can 
ever  make  such  advances  in  this  or  any  other  of  the  higher  depart- 
ments of  science  as  can  entitle  him  to  form  an  independent  opinion 
on  any  subject  of  discussion  within  their  range. — HEKSCHEL,  J. 

Outlines  of  Astronomy,  Introduction,  sect.  7. 

1564.  The  long  series  of  connected  truths  which  compose  the 
science  of  astronomy,  have  been  evolved  from  the  appearances 
and  observations  by  calculation,  and  a  process  of  reasoning 
entirely  geometrical.     It  was  not  without  reason  that  Plato 
called  geometry  and  arithmetic  the  wings  of  astronomy;  for 
it  is  only  by  means  of  these  two  sciences  that  we  can  give  a 
rational  account  of  any  of  the  appearances,  or  connect  any  fact 
with  theory,  or  even  render  a  single  observation  available  to  the 
most  common  astronomical  purpose.    It  is  by  geometry  that 
we  are  enabled  to  reason  our  way  up  through  the  apparent 
motions  to  the  real  orbits  of  the  planets,  and  to  assign  their 
positions,  magnitudes  and  eccentricities.    And  it  is  by  applica- 
tion of  geometry — a  sublime  geometry,  indeed,  invented  for  the 
purpose — to  the  general  laws  of  mechanics,  that  we  demon- 
strate the  law  of  gravitation,  trace  it  through  its  remotest 
effects  on  the  different  planets,  and,  comparing  these  effects 
with  what  we  observe,  determine  the  densities  and  weights  of 
the  minutest  bodies  belonging  to   the  system.     The  whole 
science  of  astronomy  is  in  fact  a  tissue  of  geometrical  reason- 
ing, applied  to  the  data  of  observation;  and  it  is  from  this  cir- 
cumstance that  it  derives  its  peculiar  character  of  precision  and 
certainty.    To  disconnect  it  from  geometry,  therefore,  and  to 
substitute  familiar  illustrations  and  vague  description  for  close 
and  logical  reasoning,  is  to  deprive  it  of  its  principal  advantages, 


MATHEMATICS   AND    SCIENCE  245 

and  to  reduce  it  to  the  condition  of  an  ordinary  province  of 
natural  history. 

Edinburgh  Review,  Vol.  58  (1833-1834),  p.  168. 

1565.  But  geometry  is  not  only  the  instrument  of  astronomi- 
cal investigation,  and  the  bond  by  which  the  truths  are  en- 
chained together, — it  is  also  the  instrument  of  explanation, 
affording,  by  the  peculiar  brevity  and  perspicuity  of  its  technical 
processes,  not  only  aid  to  the  learner,  but  also  such  facilities  to 
the  teacher  as  he  will  find  it  very  difficult  to  supply,  if  he  volun- 
tarily undertakes  to  forego  its  assistance.    Few  undertakings, 
indeed,  are  attended  with  greater  difficulty  than  that  of  at- 
tempting to  exhibit  the  connecting  links  of  a  chain  of  mathe- 
matical reasoning,  when  we  lay  aside  the  technical  symbols  and 
notation  which  relieve  the  memory,  and  speak  at  once  to  the 
eyes  and  the  understanding:  .  .  . 

Edinburgh  Review,  Vol.  58  (1833-1834),  p.  169. 

1566.  With  an  ordinary  acquaintance  of  trigonometry,  and 
the  simplest  elements  of  algebra,  one  may  take  up  any  well- 
written  treatise  on  plane  astronomy,  and  work  his  way  through 
it,  from  beginning  to  end,  with  perfect  ease;  and  he  will  ac- 
quire, in  the  course  of  his  progress,  from  the  mere  examples  put 
before  him,  an  infinitely  more  correct  and  precise  idea  of  as- 
tronomical methods  and  theories,  than  he  could  obtain  in  a 
lifetime  from  the  most  eloquent  general  descriptions  that  ever 
were  written.    At  the  same  time  he  will  be  strengthening  him- 
self for  farther  advances,  and  accustoming  his  mind  to  habits  of 
close  comparison  and  rigid  demonstration,  which  are  of  in- 
finitely more  importance  than  the  acquisition  of  stores  of 

undigested  facts. 

Edinburgh    Review,     Vol.    58    (1833-1834), 
p.  170. 

1567.  While  the  telescope  serves  as  a  means  of  penetrating 
space,  and  of  bringing  its  remotest  regions  nearer  us,  mathe- 
matics, by  inductive  reasoning,  have*  led  us  onwards  to  the 
remotest  regions  of  heaven,  and  brought  a  portion  of  them 
within  the  range  of  our  possibilities;  nay,  in  our  own  times — so 
propitious  to  the  extension  of  knowledge — the  application  of 


246  MEMORABILIA   MATHEMATICA 

all  the  elements  yielded  by  the  present  conditions  of  astronomy 
has  even  revealed  to  the  intellectual  eyes  a  heavenly  body,  and 
assigned  to  it  its  place,  orbit,  mass,  before  a  single  telescope 
has  been  directed  towards  it. — HUMBOLDT,  A. 

Cosmos  [Otte],  Vol.  2,  part  2,  sect.  8. 

1568.  Mighty  are  numbers,  joined  with  art  resistless. 

EURIPIDES. 
Hecuba,  Line  884. 

1569.  No  single  instrument  of  youthful  education  has  such 
mighty  power,  both  as  regards  domestic  economy  and  politics, 
and  in  the  arts,  as  the  study  of  arithmetic.    Above  all,  arith- 
metic stirs  up  him  who  is  by  nature  sleepy  and  dull,  and  makes 
him  quick  to  learn,  retentive,  shrewd,  and  aided  by  art  divine 
he  makes  progress  quite  beyond  his  natural  powers. — PLATO. 

Laws  [Jowett,]  Bk.  5,  p.  747. 

1570.  For  all  the  higher  arts  of  construction  some  acquaint- 
ance with  mathematics  is  indispensable.    The  village  carpenter, 
who,  lacking  rational  instruction,  lays  out  his  work  by  empirical 
rules  learned  in  his  apprenticeship,  equally  with  the  builder  of  a 
Britannia  Bridge,  makes  hourly  reference  to  the  laws  of  quanti- 
tative relations.     The  surveyor  on  whose  survey  the  land  is 
purchased;  the  architect  in  designing  a  mansion  to  be  built  on  it; 
the  builder  in  preparing  his  estimates;  his  foreman  in  laying  out 
the  foundations;  the  masons  in  cutting  the  stones;  and  the 
various  artisans  who  put  up  the  fittings;  are  all  guided  by 
geometrical  truths.    Railway-making  is  regulated  from  begin- 
ning to  end  by  mathematics:  alike  in  the  preparation  of  plans 
and  sections;  in  staking  out  the  lines;  in  the  mensuration  of 
cuttings  and  embankments;  hi  the  designing,  estimating,  and 
building  of  bridges,  culverts,  viaducts,  tunnels,  stations.    And 
similarly  with  the  harbors,  docks,  piers,  and  various  engineering 
and  architectural  works  that  fringe  the  coasts  and  overspread 
the  face  of  the  country,  as  well  as  the  mines  that  run  under- 
neath it.    Out  of  geometry,  too,  as  applied  to  astronomy,  the 
art  of  navigation  has  grown;  and  so,  by  this  science,  has  been 
made  possible  that  enormous  foreign  commerce  which  supports 
a  large  part  of  our  population,  and  supplies  us  with  many 


MATHEMATICS   AND    SCIENCE  247 

necessaries  and  most  of  our  luxuries.  And  nowadays  even  the 
farmer,  for  the  correct  laying  out  of  his  drains,  has  recourse  to 
the  level — that  is,  to  geometrical  principles. 

SPENCER,  HERBERT. 

Education,  chap.  1. 

1571.  [Arithmetic]  is  another  of  the  great  master-keys  of 
life.    With  it  the  astronomer  opens  the  depths  of  the  heavens; 
the  engineer,  the  gates  of  the  mountains;  the  navigator,  the 
pathways  of  the  deep.     The  skillful  arrangement,  the  rapid 
handling  of  figures,  is  a  perfect  magician's  wand.    The  mighty 
commerce  of  the  United  States,  foreign  and  domestic,  passes 
through  the  books  kept  by  some  thousands  of  diligent  and 
faithful  clerks.     Eight  hundred  bookkeepers,  in  the  Bank  of 
England,  strike  the  monetary  balance  of  half  the  civilized  world. 
Their  skill  and  accuracy  in  applying  the  common  rules  of 
arithmetic  are  as  important  as  the  enterprise  and  capital  of  the 
merchant,  or  the  industry  and  courage  of  the  navigator.    I  look 
upon  a  well-kept  ledger  with  something  of  the  pleasure  with 
which  I  gaze  on  a  picture  or  a  statue.    It  is  a  beautiful  work  of 
art. — EVERETT,  EDWARD. 

Orations  and  Speeches  (Boston,  1870},  Vol.  S, 
p.  47. 

1572.  [Mathematics]  is  the  fruitful  Parent  of,  I  had  almost 
said  all,  Arts,  the  unshaken  Foundation  of  Sciences,  and  the 
plentiful  Fountain  of  Advantage  to  Human  Affairs.    In  which 
last  Respect,  we  may  be  said  to  receive  from  the  Mathematics, 
the  principal  Delights  of  Life,  Securities  of  Health,  Increase  of 
Fortune,  and  Conveniences  of  Labour:  That  we  dwell  elegantly 
and  commodiously,  build  decent  Houses  for  ourselves,  erect 
stately  Temples  to  God,  and  leave  wonderful  Monuments  to 
Posterity:  That  we  are  protected  by  those  Rampires  from  the 
Incursions  of  the  Enemy;  rightly  use  Arms,  skillfully  range  an 
Army,  and  manage  War  by  Art,  and  not  by  the  Madness  of 
wild  Beasts:  That  we  have  safe  Traffick  through  the  deceitful 
Billows,  pass  in  a  direct  Road  through  the  tractless  Ways  of 
the  Sea,  and  come  to  the  designed  Ports  by  the  uncertain 
Impulse  of  the  Winds:  That  we  rightly  cast  up  our  Accounts,  do 
Business  expeditiously,  dispose,  tabulate,  and  calculate  scat- 


248  MEMORABILIA   MATHEMATICA 

tered  Ranks  of  Numbers,  and  easily  compute  them,  though 
expressive  of  huge  Heaps  of  Sand,  nay  immense  Hills  of  Atoms : 
That  we  make  pacifick  Separations  of  the  Bounds  of  Lands, 
examine  the  Momenta  of  Weights  hi  an  equal  Balance,  and 
distribute  every  one  his  own  by  a  just  Measure:  That  with  a 
light  Touch  we  thrust  forward  vast  Bodies  which  way  we  will, 
and  stop  a  huge  Resistance  with  a  very  small  Force:  That  we 
accurately  delineate  the  Face  of  this  Earthly  Orb,  and  subject 
the  Oeconomy  of  the  Universe  to  our  Sight :  That  we  aptly 
digest  the  flowing  Series  of  Time,  distinguish  what  is  acted  by 
due  Intervals,  rightly  account  and  discern  the  various  Returns 
of  the  Seasons,  the  stated  Periods  of  Years  and  Months,  the 
alternate  Increments  of  Days  and  Nights,  the  doubtful  Limits 
of  Light  and  Shadow,  and  the  exact  Differences  of  Hours  and 
Minutes :  That  we  derive  the  subtle  Virtue  of  the  Solar  Rays  to 
our  Uses,  infinitely  extend  the  Sphere  of  Sight,  enlarge  the  near 
Appearances  of  Things,  bring  to  Hand  Things  remote,  discover 
Things  hidden,  search  Nature  out  of  her  Concealments,  and 
unfold  her  dark  Mysteries:  That  we  delight  our  Eyes  with 
beautiful  Images,  cunningly  imitate  the  Devices  and  portray 
the  Works  of  Nature;  imitate  did  I  say?  nay  excel,  while  we 
form  to  ourselves  Things  not  in  being,  exhibit  Things  absent, 
and  represent  Things  past:  That  we  recreate  our  Minds  and 
delight  our  Ears  with  melodious  Sounds,  attemperate  the 
inconstant  Undulations  of  the  Air  to  musical  Tunes,  add  a 
pleasant  Voice  to  a  sapless  Log  and  draw  a  sweet  Eloquence 
from  a  rigid  Metal;  celebrate  our  Maker  with  an  harmonious 
Praise,  and  not  unaptly  imitate  the  blessed  Choirs  of  Heaven: 
That  we  approach  and  examine  the  inaccessible  Seats  of  the 
Clouds,  the  distant  Tracts  of  Land,  unfrequented  Paths  of  the 
Sea;  lofty  Tops  of  the  Mountains,  low  Bottoms  of  the  Valleys, 
and  deep  Gulphs  of  the  Ocean:  That  in  Heart  we  advance  to  the 
Saints  themselves  above,  yea  draw  them  to  us,  scale  the  etherial 
Towers,  freely  range  through  the  celestial  Fields,  measure  the 
Magnitudes,  and  determine  the  Interstices  of  the  Stars,  pre- 
scribe inviolable  Laws  to  the  Heavens  themselves,  and  confine 
the  wandering  Circuits  of  the  Stars  within  fixed  Bounds: 
Lastly,  that  we  comprehend  the  vast  Fabrick  of  the  Universe, 
admire  and  contemplate  the  wonderful  Beauty  of  the  Divine 


MATHEMATICS   AND    SCIENCE  249 

Workmanship,  and  to  learn  the  incredible  Force  and  Sagacity 
of  our  own  Minds,  by  certain  Experiments,  and  to  acknowledge 
the  Blessings  of  Heaven  with  pious  Affection. 

BARROW,  ISAAC. 

Mathematical  Lectures  (London,  1784),  PP-  27-80. 

1573.  Analytical   and   graphical   treatment  of  statistics  is 
employed  by  the  economist,  the  philanthropist,  the  business 
expert,  the  actuary,  and  even  the  physician,  with  the  most 
surprisingly  valuable  results;  while  symbolic  language  involving 
mathematical  methods  has  become  a  part  of  wellnigh  every 
large  business.    The  handling  of  pig-iron  does  not  seem  to  offer 
any  opportunity  for  mathematical  application.    Yet  graphical 
and   analytical   treatment  of  the   data  from  long-continued 
experiments  with  this  material  at  Bethlehem,  Pennsylvania, 
resulted  in  the  discovery  of  the  law  that  fatigue  varied  in  propor- 
tion to  a  certain  relation  between  the  load  and  the  periods  of 
rest.     Practical  application  of  this  law  increased  the  amount 
handled  by  each  man  from  twelve  and  a  half  to  forty-seven  tons 
per  day.     Such  study  would  have  been  impossible  without 
preliminary  acquaintance  with  the  simple  invariable  elements 
of  mathematics. — KARPINSKY,  L. 

High    School    Education    (New    York,    1912}, 
chap.  6,  p.  134. 

1574.  They    [computation  and  arithmetic]  belong   then,   it 
seems,  to  the  branches  of  learning  which  we  are  now  investi- 
gating;— for  a  military  man  must  necessarily  learn  them  with  a 
view  to  the  marshalling  of  his  troops,  and  so  must  a  philosopher 
with  the  view  of  understanding  real  being,  after  having  emerged 
from  the  unstable  condition  of  becoming,  or  else  he  can  never 
become  an  apt  reasoner. 

That  is  the  fact  he  replied. 

But  the  guardian  of  ours  happens  to  be  both  a  military  man 
and  a  philosopher. 

Unquestionably  so. 

It  would  be  proper  then,  Glaucon,  to  lay  down  laws  for  this 
branch  of  science  and  persuade  those  about  to  engage  hi  the  most 
important  state-matters  to  apply  themselves  to  computation, 


250  MEMORABILIA   MATHEMATICA 

and  study  it,  not  in  the  common  vulgar  fashion,  but  with  the 
view  of  arriving  at  the  contemplation  of  the  nature  of  numbers 
by  the  intellect  itself, — not  for  the  sake  of  buying  and  selling  as 
anxious  merchants  and  retailers,  but  for  war  also,  and  that  the 
soul  may  acquire  a  facility  hi  turning  itself  from  what  is  in  the 
course  of  generation  to  truth  and  real  being. — PLATO. 

Republic  [Davis],  Bk.  7,  p.  525. 

1575.  The  scientific  part  of  Arithmetic  and  Geometry  would 
be  of  more  use  for  regulating  the  thoughts  and  opinions  of  men 
than  all  the  great  advantage  which  Society  receives  from  the 
general  application  of  them:  and  this  use  cannot  be  spread 
through  the  Society  by  the  practice;  for  the  Practitioners, 
however  dextrous,  have  no  more  knowledge  of  the  Science  than 
the  very  instruments  with  which  they  work.    They  have  taken 
up  the  Rules  as  they  found  them  delivered  down  to  them  by 
scientific  men,  without  the  least  inquiry  after  the  Principles 
from  which  they  are  derived:  and  the  more  accurate  the  Rules, 
the  less  occasion  there  is  for  inquiring  after  the  Principles,  and 
consequently,  the  more  difficult  it  is  to  make  them  turn  their 
attention  to  the  First  Principles;  and,  therefore,  a  Nation  ought 
to  have  both  Scientific  and  Practical  Mathematicians. 

WILLIAMSON,  JAMES. 

Elements  of  Euclid  with  Dissertations  (Oxford, 
1781). 

1576.  Where  there  is  nothing  to  measure  there  is  nothing  to 
calculate,   hence  it  is  impossible  to  employ  mathematics  in 
psychological  investigations.     Thus  runs  the  syllogism  com- 
pounded of  an  adherence  to  usage  and  an  apparent  truth.    As  to 
the  latter,  it  is  wholly  untrue  that  we  may  calculate  only  where 
we  have  measured.     Exactly  the  opposite  is  true.     Every 
hypothetically  assumed  law  of  quantitative  combination,  even 
such  as  is  recognized  as  invalid,  is  subject  to  calculation;  and  in 
case  of  deeply  hidden  but  important  matters  it  is  imperative 
to  try  on  hypotheses  and  to  subject  the  consequences  which 
flow  from  them  to  precise  computation  until  it  is  found  which 
one  of  the  various  hypotheses  coincides  with  experience.    Thus 
the   ancient  astronomers   tried   eccentric  circles,   and  Kepler 


MATHEMATICS   AND    SCIENCE  251 

tried  the  ellipse  to  account  for  the  motion  of  the  planets,  the 
latter  also  compared  the  squares  of  the  times  of  revolution  with 
the  cubes  of  the  mean  distances  before  he  discovered  their 
agreement.  In  like  manner  Newton  tried  whether  a  gravitation, 
varying  inversely  as  the  square  of  the  distance,  sufficed  to  keep 
the  moon  in  its  orbit  about  the  earth;  if  this  supposition  had 
failed  him,  he  would  have  tried  some  other  power  of  the  distance, 
as  the  fourth  or  fifth,  and  deduced  the  corresponding  conse- 
quences to  compare  them  with  the  observations.  Just  this  is 
the  greatest  benefit  of  mathematics,  that  it  enables  us  to  sur- 
vey the  possibilities  whose  range  includes  the  actual,  long  before 
we  have  adequate  definite  experience;  this  makes  it  possible  to 
employ  very  incomplete  indications  of  experience  to  avoid  at 
least  the  crudest  errors.  Long  before  the  transit  of  Venus  was 
employed  in  the  determination  of  the  sun's  parallax,  it  was 
attempted  to  determine  the  instant  at  which  the  sun  illumines 
exactly  one-half  of  the  moon's  disk,  in  order  to  compute  the 
sun's  distance  from  the  known  distance  of  the  moon  from  the 
earth.  This  was  not  possible,  for,  owing  to  psychological 
reasons,  our  method  of  measuring  time  is  too  crude  to  give  us  the 
desired  instant  with  sufficient  accuracy;  yet  the  attempt  gave 
us  the  knowledge  that  the  sun's  distance  from  us  is  at  least 
several  hundred  times  as  great  as  that  of  the  moon.  This 
illustration  shows  clearly  that  even  a  very  imperfect  estimate  of 
a  magnitude  in  a  case  where  no  precise  observation  is  possible, 
may  become  very  instructive,  if  we  know  how  to  exploit  it.  Was 
it  necessary  to  know  the  scale  of  our  solar  system  in  order  to  learn 
of  its  order  in  general?  Or,  taking  an  illustration  from  another 
field,  was  it  impossible  to  investigate  the  laws  of  motion  until 
it  was  known  exactly  how  far  a  body  falls  in  a  second  at  some 
definite  place?  Not  at  all.  Such  determinations  of  fundamental 
measures  are  in  themselves  exceedingly  difficult,  but  fortunately, 
such  investigations  form  a  class  of  their  own;  our  knowledge  of 
fundamental  laws  does  not  need  to  wait  on  these.  To  be  sure, 
computation  invites  measurement,  and  every  easily  observed 
regularity  of  certain  magnitudes  is  an  incentive  to  mathematical 
investigation. — HERBART,  J.  F. 

Werke  [Kehrbach],  (Langensalza,  1890),  Bd.  5, 

p.  97. 


252  MEMORABILIA   MATHEMATICA 

1677.  Those  who  pass  for  naturalists,  have,  for  the  most 
part,  been  very  little,  or  not  at  all,  versed  in  mathematicks,  if 
not  also  jealous  of  them. — BOYLE,  ROBERT. 

Works  (London,  1772},  Vol.  3,  p.  426. 

1578.  However  hurtful  may  have  been  the  incursions  of  the 
geometers,  direct  and  indirect,  into  a  domain  which  it  is  not  for 
them  to  cultivate,  the  physiologists  are  not  the  less  wrong  in 
turning  away  from  mathematics  altogether.    It  is  not  only  that 
without  mathematics  they  could  not  receive  their  due  prelimi- 
nary training  in  the  intervening  sciences :  it  is  further  necessary 
for  them  to  have  geometrical  and  mechanical  knowledge,  to 
understand  the  structure  and  the  play  of  the  complex  apparatus 
of  the  living,  and  especially  the  animal  organism.     Animal 
mechanics,  statical  and  dynamical,  must  be  unintelligible  to 
those  who  are  ignorant  of  the  general  laws  of  rational  mechanics. 
The  laws  of  equilibrium  and  motion  are  .  .  .  absolutely  univer- 
sal hi  their  action,  depending  wholly  on  the  energy,  and  not  at 
all  on  the  nature  of  the  forces  considered :  and  the  only  difficulty 
is  in  their  numerical  application  in  cases  of  complexity.    Thus, 
discarding  all  idea  of  a  numerical  application  in  biology,  we 
perceive  that  the  general  theorems  of  statics  and  dynamics  must 
be  steadily  verified  in  the  mechanism  of  living  bodies,  on  the 
rational  study  of  which  they  cast  an  indispensable  light.    The 
highest  orders  of  animals  act  in  repose  and  motion,  like  any 
other  mechanical  apparatus  of  a  similar  complexity,  with  the  one 
difference  of  the  mover,  which  has  no  power  to  alter  the  laws  of 
motion  and  equilibrium.    The  participation  of  rational  mechan- 
ics in  positive  biology  is  thus  evident.     Mechanics  cannot 
dispense  with  geometry;  and  beside,  we  see  how  anatomical  and 
physiological  speculations  involve  considerations  of  form  and 
position,  and  require  a  familiar  knowledge  of  the  principal 
geometrical  laws  which  may  cast  light  upon  these  complex 
relations. — COMTE,  A. 

Positive  Philosophy  [Martineau],  Bk.  5,  chap.  1. 

1579.  In  mathematics  we  find  the  primitive  source  of  ra- 
tionality; and  to  mathematics  must  the  biologists  resort  for 
means  to  carry  on  their  researches. — COMTE,  A. 

Positive  Philosophy  [Martineau],  Bk.  5,  chap.  1. 


MATHEMATICS   AND    SCIENCE  253 

1580.  In  this  school  [of  mathematics]  must  they  [biologists] 
learn  familiarly  the  real  characters  and  conditions  of  scientific 
evidence,  in  order  to  transfer  it  afterwards  to  the  province  of 
their  own  theories.  The  study  of  it  here,  in  the  most  simple  and 
perfect  cases,  is  the  only  sound  preparation  for  its  recognition  in 
the  most  complex. 

The  study  is  equally  necessary  for  the  formation  of  intellectual 
habits;  for  obtaining  an  aptitude  in  forming  and  sustaining 
positive  abstractions,  without  which  the  comparative  method 
cannot  be  used  in  either  anatomy  or  physiology.  The  abstrac- 
tion which  is  to  be  the  standard  of  comparison  must  be  first 
clearly  formed,  and  then  steadily  maintained  in  its  integrity,  or 
the  analysis  becomes  abortive:  and  this  is  so  completely  in  the 
spirit  of  mathematical  combinations,  that  practice  in  them  is 
the  best  preparation  for  it.  A  student  who  cannot  accomplish 
the  process  in  the  more  simple  case  may  be  assured  that  he  is  not 
qualified  for  the  higher  order  of  biological  researches,  and  must 
be  satisfied  with  the  humbler  office  of  collecting  materials  for 
the  use  of  minds  of  another  order.  Hence  arises  another  use  of 
mathematical  training; — that  of  testing  and  classifying  minds, 
as  well  as  preparing  and  guiding  them.  Probably  as  much  good 
would  be  done  by  excluding  the  students  who  only  encumber  the 
science  by  aimless  and  desultory  inquiries,  as  by  fitly  instituting 
those  who  can  better  fulfill  its  conditions. — COMTE,  A. 

Positive    Philosophy    [Martineau],    Bk.    5, 
chap.  1. 

1581.  There  seems  no  sufficient  reason  why  the  use  of  scientific 
fictions,  so  common  in  the  hands  of  geometers,  should  not  be 
introduced  into  biology,  if  systematically  employed,  and 
adopted  with  sufficient  sobriety.  In  mathematical  studies, 
great  advantages  have  arisen  from  imagining  a  series  of  hypo- 
thetical cases,  the  consideration  of  which,  though  artificial,  may 
aid  the  clearing  up  of  the  real  subject,  or  its  fundamental 
elaboration.  This  art  is  usually  confounded  with  that  of 
hypotheses;  but  it  is  entirely  different;  inasmuch  as  in  the  latter 
case  the  solution  alone  is  imaginary;  whereas  in  the  former,  the 
problem  itself  is  radically  ideal.  Its  use  can  never  be  in  biology 
comparable  to  what  it  is  in  mathematics :  but  it  seems  to  me  that 


254  MEMORABILIA    MATHEMATICA 

the  abstract  character  of  the  higher  conceptions  of  comparative 
biology  renders  them  susceptible  of  such  treatment.  The  proc- 
ess will  be  to  intercalate,  among  different  known  organisms, 
certain  purely  fictitious  organisms,  so  imagined  as  to  facilitate 
their  comparison,  by  rendering  the  biological  series  more  homo- 
geneous and  continuous:  and  it  might  be  that  several  might 
hereafter  meet  with  more  or  less  ot  a  realization  among  or- 
ganisms hitherto  unexplored.  It  may  be  possible,  in  the  present 
state  of  our  knowledge  of  living  bodies,  to  conceive  of  a  new 
organism  capable  of  fulfilling  certain  given  conditions  of  exist- 
ence. However  that  may  be,  the  collocation  of  real  cases  with 
well-imagined  ones,  after  the  manner  of  geometers,  will  doubt- 
less be  practised  hereafter,  to  complete  the  general  laws  of 
comparative  anatomy  and  physiology,  and  possibly  to  antici- 
pate occasionally  the  direct  exploration.  Even  now,  the  rational 
use  of  such  an  artifice  might  greatly  simplify  and  clear  up  the 
ordinary  system  of  biological  instruction.  But  it  is  only  the 
highest  order  of  investigators  who  can  be  trusted  with  it. 
Whenever  it  is  adopted,  it  will  constitute  another  ground  of 
relation  between  biology  and  mathematics. — COMTE,  A. 

Positive    Philosophy    [Martineau],     Bk.    5, 
chap.  1. 

1582.  I  think  it  may  safely  enough  be  affirmed,  that  he,  that 
is  not  so  much  as  indifferently  skilled  in  mathematicks,  can 
hardly  be  more  than  indifferently  skilled  in  the  fundamental 
principles  of  physiology. — BOYLE,  ROBERT. 

Works  (London,  1772},  Vol.  8,  p.  480. 

1583.  It  is  not  only  possible  but  necessaiy  that  mathe- 
matics be  applied  to  psychology;  the  reason  for  this  necessity 
lies  briefly  in  this :  that  by  no  other  means  can  be  reached  that 
which  is  the  ultimate  aim  of  all  speculation,  namely  conviction. 

HERBART,  J.  F. 

Werke  [Kehrbach],  (Langensalza,  1890},  Bd.  5, 
p.  104. 

1584.  All  more  definite  knowledge  must  start  with  computa- 
tion; and  this  is  of  most  important  consequences  not  only  for 


MATHEMATICS   AND    SCIENCE  255 

the  theory  of  memory,  of  imagination,  of  understanding,  but  as 
well  for  the  doctrine  of  sensations,  of  desires,  and  affections. 

HERBART,  J.  F. 

Werke  [Kehrbach],  (Langensalza,  1890},  Bd.  5, 

p.  103. 

1585.  In  the  near  future  mathematics  will  play  an  important 
part  in  medicine:  already  there  are  increasing  indications  that 
physiology,  descriptive  anatomy,  pathology  and  therapeutics 
cannot  escape  mathematical  legitimation. — DESSOIR,  MAX. 

Westermann's  Monatsberichte,  Bd.  77,  p.  880; 
Ahrens:  Scherz  und  Ernst  in  der  Mathematik 
(Leipzig,  1904),  P-  895. 

1586.  The  social  sciences  mathematically  developed  are  to  be 
the  controlling  factors  in  civilization. — WHITE,  W.  F. 

A  Scrap-book  of  Elementary  Mathematics 
(Chicago,  1908),  p.  208. 

1587.  It  is  clear  that  this  education  [referring  to  education 
preparatory  to  the  science  of  sociology]  must  rest  on  a  basis  of 
mathematical  philosophy,  even  apart  from  the  necessity  of 
mathematics  to  the  study  of  inorganic  philosophy.    It  is  only 
in  the  region  of  mathematics  that  sociologists,  or  anybody  else, 
can  obtain  a  true  sense  of  scientific  evidence,  and  form  the 
habit  of  rational  and  decisive  argumentation;  can,  in  short, 
learn  to  fulfill  the  logical  conditions  of  all  positive  speculation,  by 
studying  universal  positivism  at  its  source.     This  training, 
obtained  and  employed  with  the  more  care  on  account  of  the 
eminent  difficulty  of  social  science,  is  what  sociologists  have  to 
seek  in  mathematics. — COMTE,  A. 

Positive    Philosophy    [Martineau],    Bk.    6, 
chap.  4- 

1588.  It  is  clear  that  the  individual  as  a  social  unit  and  the 
state  as  a  social  aggregate  require  a  certain  modicum  of  mathe- 
matics, some  arithmetic  and  algebra,  to  conduct  their  affairs. 
Under  this  head  would  fall  the  theory  of  interest,  simple  and 
compound,  matters  of  discount  and  amortization,  and,  if  lot- 
teries hold  a  prominent  place  in  raising  moneys,  as  in  some 
states,  questions  of  probability  must  be  added.    As  the  state 


256  MEMORABILIA    MATHEMATICA 

becomes  more  highly  organized  and  more  interested  in  the 
scientific  analysis  of  its  life,  there  appears  an  urgent  necessity 
for  various  statistical  information,  and  this  can  be  properly 
obtained,  reduced,  correlated,  and  interpreted  only  when  the 
guiding  spirit  in  the  work  have  the  necessary  mathematical 
training  in  the  theory  of  statistics.  (Figures  may  not  lie,  but 
statistics  compiled  unscientifically  and  analyzed  incompetently 
are  almost  sure  to  be  misleading,  and  when  this  condition  is 
unnecessarily  chronic  the  so-called  statisticians  may  well  be 
called  liars.)  The  dependence  of  insurance  of  various  kinds  on 
statistical  information  and  the  very  great  place  which  insurance 
occupies  in  the  modern  state,  albeit  often  controlled  by  private 
corporations  instead  of  by  the  government,  makes  the  theories  of 
paramount  importance  to  our  social  life. — WILSON,  E.  B. 

Bulletin    American    Mathematical    Society, 

Vol.  18  (1912],  p.  463. 

1689.  The  theory  of  probabilities  and  the  theory  of  errors 
now  constitute  a  formidable  body  of  knowledge  of  great  mathe- 
matical interest  and  of  great  practical  importance.  Though 
developed  largely  through  the  applications  to  the  more  precise 
sciences  of  astronomy,  geodesy,  and  physics,  their  range  of 
applicability  extends  to  all  the  sciences;  and  they  are  plainly 
destined  to  play  an  increasingly  important  role  in  the  develop- 
ment and  hi  the  applications  of  the  sciences  of  the  future.  Hence 
their  study  is  not  only  a  commendable  element  in  a  liberal 
education,  but  some  knowledge  of  them  is  essential  to  a  correct 
understanding  of  daily  events. — WOODWARD,  R.  S. 

Probability  and  Theory  of  Errors  (New  York, 
1906),  Preface. 

1590.  It  was  not  to  be  anticipated  that  a  new  science  [the 
science  of  probabilities]  which  took  its  rise  in  games  of  chance, 
and  which  had  long  to  encounter  an  obloquy,  hardly  yet  extinct, 
due  to  the  prevailing  idea  that  its  only  end  was  to  facilitate  and 
encourage  the  calculations  of  gamblers,  could  ever  have  attained 
its  present  status — that  its  aid  should  be  called  for  in  every 
department  of  natural  science,  both  to  assist  in  discovery,  which 
it  has  repeatedly  done  (even  in  pure  mathematics),  to  minimize 
the  unavoidable  errors  of  observation,  and  to  detect  the  presence 


MATHEMATICS  AND    SCIENCE  257 

of  causes  as  revealed  by  observed  events.  Nor  are  commercial 
and  other  practical  interests  of  life  less  indebted  to  it:  wherever 
the  future  has  to  be  forecasted,  risk  to  be  provided  against,  or 
the  true  lessons  to  be  deduced  from  statistics,  it  corrects  for  us 
the  rough  conjectures  of  common  sense,  and  decides  which 
course  is  really,  according  to  the  lights  of  which  we  are  in  posses- 
sion, the  wisest  for  us  to  pursue. — CROFTON,  M.  W. 

Encyclopedia  Britannica,  9th  Edition;  Article 
"  Probability." 

1591.  The  calculus  of  probabilities,  when  confined  within 
just  limits,  ought  to  interest,  in  an  equal  degree,  the  mathe- 
matician, the  experimentalist,  and  the  statesman.    From  the 
time  when  Pascal  and  Fermat  established  its  first  principles,  it 
has  rendered,  and  continues  daily  to  render,  services  of  the  most 
eminent  kind.    It  is  the  calculus  of  probabilities,  which,  after 
having  suggested  the  best  arrangements  of  the  tables  of  popula- 
tion and  mortality,  teaches  us  to  deduce  from  those  numbers,  in 
general  so  erroneously  interpreted,  conclusions  of  a  precise  and 
useful  character;  it  is  the  calculus  of  probabilities  which  alone 
can  regulate  justly  the  premiums  to  be  paid  for  assurances; 
the  reserve  funds  for  the  disbursements  of  pensions,  annuities, 
discounts,  etc.    It  is  under  its  influence  that  lotteries  and  other 
shameful  snares  cunningly  laid  for  avarice  and  ignorance  have 
definitely  disappeared. — ARAGO. 

Eulogy   on   Laplace    [Baden-Powell],   Smith- 
sonian Report,  1874,  P- 164- 

1592.  Men  were  surprised  to  hear  that  not  only  births,  deaths, 
and  marriages,  but  the  decisions  of  tribunals,  the  results  of 
popular  elections,  the  influence  of  punishments  in  checking 
crime,  the  comparative  values  of  medical  remedies,  the  probable 
limits  of  error  in  numerical  results  in  every  department  of 
physical  inquiry,  the  detection  of  causes,  physical,  social,  and 
moral,  nay,  even  the  weight  of  evidence  and  the  validity  of 
logical  argument,  might  come  to  be  surveyed  with  the  lynx- 
eyed  scrutiny  of  a  dispassionate  analysis. — HERSCHEL,  J. 

Quoted  in  Encyclopedia  Britannica,  9th  Edi- 
tion; Article  "Probability." 


258  MEMORABILIA   MATHEMATICA 

1593.  If  economists  expect  of  the  application  of  the  mathe- 
matical method  any  extensive  concrete  numerical  results,  and 
it  is  to  be  feared  that  like  other  non-mathematicians  all  too 
many  of  them  think  of  mathematics  as  merely  an  arithmetical 
science,  they  are  bound  to  be  disappointed  and  to  find  a  paucity 
of  results  hi  the  works  of  the  few  of  their  colleagues  who  use  that 
method.    But  they  should  rather  learn,  as  the  mathematicians 
among  them  know  full  well,  that  mathematics  is  much  broader, 
that  it  has  an  abstract  quantitative  (or  even  qualitative)  side, 
that  it  deals  with  relations  as  well  as  numbers,  .  .  . 

WILSON,  E.  B. 

Bulktin    American    Mathematical    Society, 
Vol.  18  (1912},  p.  464- 

1594.  The  effort  of  the  economist  is  to  see,  to  picture  the 
inter-play  of  economic  elements.    The  more  clearly  cut  these 
elements  appear  in  his  vision,  the  better;  the  more  elements  he 
can  grasp  and  hold  in  his  mind  at  once,  the  better.     The  eco- 
nomic world  is  a  misty  region.    The  first  explorers  used  unaided 
vision.    Mathematics  is  the  lantern  by  which  what  before  was 
dimly  visible  now  looms  up  hi  firm,  bold  outlines.     The  old 
phantasmagoria  disappear.    We  see  better.    We  also  see  further. 

FISHER,  IRVING. 

Transactions  of  Connecticut  Academy,  Vol.  9 
(1892),  p.  119. 

1595.  In  the  great  inquiries  of  the  moral  and  social  sci- 
ences .  .  .  mathematics  (I  always  mean  Applied  Mathemat- 
ics) affords  the  only  sufficient  type  of  deductive  art.     Up 
to  this  tune,  I  may  venture  to  say  that  no  one  ever  knew  what 
deduction  is,  as  a  means  of  investigating  the  laws  of  nature,  who 
had  not  learned  it  from  mathematics,  nor  can  any  one  hope  to 
understand  it  thoroughly,  who  has  not,  at  some  time  in  his 
life,  known  enough  of  mathematics  to  be  familiar  with  the  in- 
strument at  work. — MILL,  J.  S. 

An  Examination  of  Sir  William  Hamilton's 
Philosophy  (London,  1878),  p.  622. 

1596.  Let  me  pass  on  to  say  a  word  or  two  about  the  teaching 
of  mathematics  as  an  academic  training  for  general  professional 


MATHEMATICS   AND    SCIENCE  259 

life.  It  has  immense  capabilities  in  that  respect.  If  you  con- 
sider how  much  of  the  effectiveness  of  an  administrator  de- 
pends upon  the  capacity  for  co-ordinating  appropriately  a 
number  of  different  ideas,  precise  accuracy  of  definition,  rigidity 
of  proof,  and  sustained  reasoning,  strict  in  every  step,  and  when 
you  consider  what  substitutes  for  these  things  nine  men  out  of 
every  ten  without  special  training  have  to  put  up  with,  it  is 
clear  that  a  man  with  a  mathematical  training  has  incalculable 
advantages. — SHAW,  W.  H. 

Perry's  Teaching  of  Mathematics  (London, 
1902),  p.  73. 

1597.  Before  you  enter  on  the  study  of  law  a  sufficient 
ground  work  must  be  laid.  .  .  .  Mathematics  and  natural 
philosophy  are  so  useful  in  the  most  familiar  occurrences  of 
life  and  are  so  peculiarly  engaging  and  delightful  as  would 
induce  everyone  to  wish  an  acquaintance  with  them.    Besides 
this,  the  faculties  of  the  mind,  like  the  members  of  a  body,  are 
strengthened  and  improved  by  exercise.    Mathematical  reason- 
ing and  deductions  are,  therefore,  a  fine  preparation  for  in- 
vestigating the  abstruse  speculations  of  the  law. 

JEFFERSON,  THOMAS. 

Quoted  in  Cajori's  Teaching  and  History  of 
Mathematics  in  the  U.  S.  (Washington,  1890), 
p.  35. 

1598.  It  has  been  observed  hi  England  of  the  study  of  law, — 
though  the  acquisition  of  the  most  difficult  parts  of  its  learn- 
ing, the  interpretation  of  laws,  the  comparison  of  authorities, 
and  the  construction  of  instruments,  would  seem  to  require 
philological  and  critical  training;  though  the  weighing  of  evi- 
dence and  the  investigation  of  probable  truth  belong  to  the 
province  of  the  moral  sciences,  and  the  peculiar  duties  of  the 
advocate  require  rhetorical  skill, — yet  that  a  large  proportion 
of  the  most  distinguished  members  of  the  profession  has  pro- 
ceeded from  the  university  (that  of  Cambridge)  most  cele- 
brated for  the  cultivation  of  mathematical  studies. 

EVERETT,  EDWARD. 

Orations  and  Speeches  (Boston,  1870),  Vol.  2, 
p.  511. 


260  MEMORABILIA   MATHEMATICA 

1599.  All  historic  science  tends  to  become  mathematical. 
Mathematical  power  is  classifying  power. — NOVALIS. 

Schriften    (Berlin,    1901),    Teil   2,    p.    192. 

1599a.  History  has  never  regarded  itself  as  a  science  of 
statistics.  It  was  the  Science  of  Vital  Energy  in  relation  with 
time;  and  of  late  this  radiating  centre  of  its  life  has  been  steadily 
tending, — together  with  every  form  of  physical  and  mechanical 
energy, — toward  mathematical  expression. — ADAM,  HENRY. 

A   Letter  to  American   Teachers  of  History 

(Washington,  1910),  p.  115. 

1599b.  Mathematics  can  be  shown  to  sustain  a  certain  rela- 
tion to  rhetoric  and  may  aid  in  deteirnining  its  laws. 

SHERMAN  L.  A. 
University  [of  Nebraska]  Studies,  Vol.  1,  p.  180. 


CHAPTER  XVI 

ARITHMETIC 

1601.  There  is  no  problem  in  all  mathematics  that  cannot  be 
solved  by  direct  counting.    But  with  the  present  implements 
of  mathematics  many  operations  can  be  performed  in  a  few 
minutes  which  without  mathematical  methods  would  take  a 

lifetime. — MACH,  ERNST. 

Popular     Scientific     Lectures     [McCormack] 
(Chicago,  1898),  p.  197. 

1602.  There  is  no  inquiry  which  is  not  finally  reducible  to  a 
question  of  Numbers;  for  there  is  none  which  may  not  be  con- 
ceived of  as  consisting  in  the  determination  of  quantities  by 
each  other,  according  to  certain  relations. — COMTE,  A 

Positive  Philosophy  [Martineau],  Bk.l,  chap.  1. 

1603.  Pythagoras  says  that  number  is  the  origin  of  all  things, 
and  certainly  the  law  of  number  is  the  key  that  unlocks  the 
secrets  of  the  universe.    But  the  law  of  number  possesses  an 
immanent  order,  which  is  at  first  sight  mystifying,  but  on  a  more 
intimate  acquaintance  we  easily  understand  it  to  be  intrinsi- 
cally necessary;  and  this  law  of  number  explains  the  wondrous 

consistency  of  the  laws  of  nature. — CARUS,  PAUL. 

Reflections  on  Magic  Squares;  Monist,  Vol.  16 
(1906),  p.  139. 

1604.  An  ancient  writer  said  that  arithmetic  and  geometry 
are  the  wings  of  mathematics;  I  believe  one  can  say  without 
speaking  metaphorically  that  these  two  sciences  are  the  founda- 
tion and  essence  of  all  the  sciences  which  deal  with  quantity. 
Not  only  are  they  the  foundation,  they  are  also,  as  it  were,  the 
capstones;  for,  whenever  a  result  has  been  arrived  at,  in  order  to 
use  that  result,  it  is  necessary  to  translate  it  into  numbers  or 
into  lines;  to  translate  it  into  numbers  requires  the  aid  of  arith- 
metic, to  translate  it  into  lines  necessitates  the  use  of  geometry. 

LAGRANGE. 

Leqons  Elementaires  sur  les  Mathematiques, 
Legon  seconde. 
261 


262  MEMORABILIA  MATHEMATICA 

1605.  It  is  number  which  regulates  everything  and  it  is 
measure  which  establishes  universal  order.  ...  A  quiet  peace, 
an  inviolable  order,  an  inflexible  security  amidst  all  change  and 
turmoil  characterize  the  world  which  mathematics  discloses  and 
whose  depths  it  unlocks. — DILLMANN,  E. 

Die    Mathematik    die    Fackeltragerin    einer 
neuen  Zeit  (Stuttgart,  1889),  p.  12. 

1G06.  Number,  the  inducer  of  philosophies, 

The  synthesis  of  letters,  .  .  . — AESCHYLUS. 

Quoted  in,  Thomson,  J.  A.,  Introduction  to 
Science,  chap.  1  (London). 

1607.  Amongst  all  the  ideas  we  have,  as  there  is  none  sug- 
gested to  the  mind  by  more  ways,  so  there  is  none  more  simple, 
than  that  of  unity,  or  one :  it  has  no  shadow  of  variety  or  compo- 
sition hi  it;  every  object  our  senses  are  employed  about;  every 
idea  in  our  understanding;  every  thought  of  our  minds,  brings 
this  idea  along  with  it.    And  therefore  it  is  the  most  intimate 
to  our  thoughts,  as  well  as  it  is,  in  its  agreement  to  all  other 
things,  the  most  universal  idea  we  have. — LOCKE,  JOHN. 

An  Essay  concerning  Human  Understanding, 
Bk.2,chap.  16,  sect.  1. 

1608.  The  simple  modes  of  number  are  of  all  other  the  most 
distinct;  every  the  least  variation,  which  is  an  unit,  making  each 
combination  as  clearly  different  from  that  which  approacheth 
nearest  to  it,  as  the  most  remote;  two  being  as  distinct  from  one, 
as  two  hundred;  and  the  idea  of  two  as  distinct  from  the  idea  of 
three,  as  the  magnitude  of  the  whole  earth  is  from  that  of  a 
mite. — LOCKE,  JOHN. 

An  Essay  concerning  Human  Understanding, 
Bk.  2,  chap.  16,  sect.  8. 

1609.  The  number  of  a  class  is  the  class  of  all  classes  similar 
to  the  given  class. — RUSSELL,  BERTRAND. 

Principles  of  Mathematics  (Cambridge,  1908), 
p.  115. 

1610.  Number  is  that  property  of  a  group  of  distinct  things 
which  remains  unchanged  during  any  change  to  which  the 


ARITHMETIC  263 

group  may  be  subjected  which  does  not  destroy  the  distinctness 

of  the  individual  things. — FINE,  H.  B. 

Number-system  of  Algebra  (Boston  and  New 
York,  1890),  p.  8. 

1611.  The  science  of  arithmetic  may  be  called  the  science  of 
exact  limitation  of  matter  and  things  in  space,  force,  and  time. 

PARKER,  F.  W. 

Talks  on  Pedagogics  (New  York,  1894),  P-  ®4- 

1612.  Arithmetic  is  the  science  of  the  Evaluation  of  Func- 

tions, 

Algebra  is  the  science  of  the  Transformation  of  Func- 
tions.— HOWISON,  G.  H. 

Journal  of  Speculative  Philosophy,    Vol.  5, 
p.  175. 

1613.  That  arithmetic  rests  on  pure  intuition  of  lime  is  not  so 
obvious  as  that  geometry  is  based  on  pure  intuition  of  space, 
but  it  may  be  readily  proved  as  follows.    All  counting  consists  in 
the  repeated  positing  of  unity;  only  in  order  to  know  how  often 
it  has  been  posited,  we  mark  it  each  time  with  a  different  word: 
these  are  the  numerals.    Now  repetition  is  possible  only  through 
succession:  but  succession  rests  on  the  immediate  intuition  of 
time,  it  is  intelligible  only  by  means  of  this  latter  concept: 
hence  counting  is  possible  only  by  means  of  time. — This  de- 
pendence of  counting  on  time  is  evidenced  by  the  fact  that  in 
all  languages  multiplication  is  expressed  by  "times"   [mal], 
that  is,  by  a  concept  of  time;  sexies,  e|a/c«,  six  fois,  six  times. 

SCHOPENHAUER,  A. 

Die  Welt  als  Vorstellung  und  Wille;  Werke 
(Frauenstaedt)  (Leipzig,  1877),  Bd.  3,  p.  89. 

1614.  The   miraculous   powers   of  modern   calculation   are 
due  to  three  inventions :  the  Arabic  Notation,  Decimal  Fractions 
and  Logarithms. — CAJORI,  F. 

History  of  Mathematics  (New  York,  1897), 
p.  161. 

1615.  The  grandest  achievement  of  the  Hindoos  and  the  one 
which,  of  all  mathematical  investigations,  has  contributed  most 


264  MEMORABILIA   MATHEMATICA 

to  the  general  progress  of  intelligence,  is  the  invention  of  the 
principle  of  position  in  writing  numbers. — CAJORI,  F. 

History  of  Mathematics  (New   York,  1897), 
p.  87. 

1616.  The  invention  of  logarithms  and  the  calculation  of  the 
earlier  tables  form  a  very  striking  episode  in  the  history  of 
exact  science,  and,  with  the  exception  of  the  Principia  of  New- 
ton, there  is  no  mathematical  work  published  hi  the  country 
which  has  produced  such  important  consequences,  or  to  which 
so  much  interest  attaches  as  to  Napier's  Descriptio. 

GLAISHER,  J.  W.  L. 
Encyclopedia  Britannica,  9th  Edition;  Article 
"Logarithms." 

1617.  All  minds  are  equally  capable  of  attaining  the  science 
of  numbers:  yet  we  find  a  prodigious  difference  in  the  powers  of 
different  men,  in  that  respect,  after  they  have  grown  up,  because 
their  minds  have  been  more  or  less  exercised  in  it. 

JOHNSON,  SAMUEL. 

Boswell's  Life  of  Johnson,  Harper's  Edition 
(1871),  Vol.  2,  p.  33. 

1618.  The  method  of  arithmetical  teaching  is  perhaps  the 
best  understood  of  any  of  the  methods  concerned  with  ele- 
mentary studies. — BAIN,  ALEXANDER. 

Education  as  a  Science  (New  York,  1898), 
p.  288. 

.1619.  What  a  benefite  that  onely  thyng  is,  to  haue  the  witte 
whetted  and  sharpened,  I  neade  not  trauell  to  declare,  sith  all 
men  confesse  it  to  be  as  greate  as  maie  be.  Excepte  any  witlesse 
persone  thinke  he  maie  bee  to  wise.  But  he  that  most  feareth 
that,  is  leaste  in  daunger  of  it.  Wherefore  to  conclude,  I  see 
moare  menne  to  acknowledge  the  benefite  of  nomber,  than  I  can 
espie  willying  to  studie,  to  attaine  the  benefites  of  it.  Many 
praise  it,  but  fewe  dooe  greatly  practise  it:  onlesse  it  bee  for 
the  vulgare  practice,  concernying  Merchaundes  trade.  Wherein 
the  desire  and  hope  of  gam,  maketh  many  willying  to  sustaine 
some  trauell.  For  aide  of  whom,  I  did  sette  forth  the  first 
parte  of  Arithmetike.  But  if  thei  knewe  how  faree  this  seconde 
parte,  doeeth  excell  the  firste  parte,  thei  would  not  accoumpte 


ARITHMETIC  265 

any  tyme  loste,  that  were  emploied  in  it.  Yea  thei  would  not 
thinke  any  tyme  well  bestowed  till  thei  had  gotten  soche  habilitie 
by  it,  that  it  might  be  their  aide  in  al  other  studies. 

RECOBDE,  ROBERT. 

Whetstone  of  Witte  (London,  1557). 

1620.  You  see  then,  my  friend,  I  observed,  that  our  real  need 
of  this  branch  of  science  [arithmetic]  is  probably  because  it 
seems  to  compel  the  soul  to  use  our  intelligence  in  the  search 
after  pure  truth. 

Aye,  remarked  he,  it  does  this  to  a  remarkable  extent. 

Have  you  ever  noticed  that  those  who  have  a  turn  for  arith- 
metic are,  with  scarcely  an  exception,  naturally  quick  in  all 
sciences;  and  that  men  of  slow  intellect,  if  they  be  trained  and 
exercised  in  this  study  .  .  .  become  invariably  quicker  than 
they  were  before? 

Exactly  so,  he  replied. 

And,  moreover,  I  think  you  will  not  easily  find  that  many 
things  give  the  learner  and  student  more  trouble  than  this. 

Of  course  not. 

On  all  these  accounts,  then,  we  must  not  omit  this  branch  of 
science,  but  those  with  the  best  of  talents  should  be  instructed 
therein. — PLATO. 

Republic  [Davis],  Bk.  7,  chap.  8. 

1621.  Arithmetic   has   a  very  great   and  elevating  effect, 
compelling  the  soul  to  reason  about  abstract  number,  and  if 
visible  or  tangible  objects  are  obtruding  upon  the  argument, 
refusing  to  be  satisfied. — PLATO. 

Republic  [Jowett],  Bk.  7,  p.  525. 

1622.  Good  arithmetic  contributes  powerfully  to  purposive 
effort,  to  concentration,  to  tenacity  of  purpose,  to  generalship, 
to  faith  in  right,  and  to  the  joy  of  achievement,  which  are  the 
elements  that  make  up  efficient  citizenship.  .  .  .  Good  arith- 
metic  exalts   thinking,   furnishes   intellectual   pleasure,    adds 
appreciably  to  love  of  right,  and  subordinates  pure  memory. 

MYERS,  GEORGE. 

Monograph  on  Arithmetic  in  Public  Educa- 
tion (Chicago),  p.  21. 


266  MEMORABILIA   MATHEMATICA 

1623.  On  the  one  side  we  may  say  that  the  purpose  of  number 
work  is  to  put  a  child  in  possession  of  the  machinery  of  calcula- 
tion; on  the  other  side  it  is  to  give  him  a  better  mastery  of  the 
world  through  a  clear  (mathematical)  insight  into  the  varied 
physical  objects  and  activities.     The  whole  world,  from  one 
point  of  view,  can  be  definitely  interpreted  and  appreciated  by 
mathematical  measurements  and  estimates.    Arithmetic  hi  the 
common  school  should  give  a  child  this  point  of  view,  the 
ability  to  see  and  estimate  things  with  a  mathematical  eye. 

McMuRRAY,  C.  A. 

Special  Method  in  Arithmetic   (New    York, 

1906),  p.  18. 

\ 

1624.  We  are  so  accustomed  to  hear  arithmetic  spoken  of  as 
one  of  the  three  fundamental  ingredients  in  all  schemes  of 
instruction,  that  it  seems  like  inquiring  too  curiously  to  ask 
why  this  should  be.    Reading,  Writing,  and  Arithmetic — these 
three  are  assumed  to  be  of  co-ordinate  rank.    Are  they  indeed 
co-ordinate,  and  if  so  on  what  grounds? 

In  this  modern  "trivium"  the  art  of  reading  is  put  first. 
Well,  there  is  no  doubt  as  to  its  right  to  the  foremost  place. 
For  reading  is  the  instrument  of  all  our  acquisition.  It  is  in- 
dispensable. There  is  not  an  hour  hi  our  lives  in  which  it  does 
not  make  a  great  difference  to  us  whether  we  can  read  or  not. 
And  the  art  of  Writing,  too;  that  is  the  instrument  of  all  com- 
munication, and  it  becomes,  hi  one  form  or  other,  useful  to  us 
every  day.  But  Counting — doing  sums, — how  often  in  life 
does  this  accomplishment  come  into  exercise?  Beyond  the 
simplest  additions,  and  the  power  to  check  the  items  of  a  bill, 
the  arithmetical  knowledge  required  of  any  well-informed  person 
in  private  life  is  very  limited.  For  all  practical  purposes, 
whatever  I  may  have  learned  at  school  of  fractions,  or  propor- 
tion, or  decimals,  is,  unless  I  happen  to  be  in  business,  far  less 
available  to  me  in  life  than  a  knowledge,  say,  of  history  of  my 
own  country,  or  the  elementary  truths  of  physics.  The  truth  is, 
that  regarded  as  practical  arts,  reading,  writing,  and  arithmetic 
have  no  right  to  be  classed  together  as  co-ordinate  elements  of 
education;  for  the  last  of  these  is  considerably  less  useful  to  the 
average  man  or  woman  not  only  than  the  other  two,  but  than 


ARITHMETIC  267 

many  others  that  might  be  named.  But  reading,  writing,  and 
such  mathematical  or  logical  exercise  as  may  be  gained  in  con- 
nection with  the  manifestation  of  numbers,  have  a  right  to  con- 
stitute the  primary  elements  of  instruction.  And  I  believe  that 
arithmetic,  if  it  deserves  the  high  place  that  it  conventionally 
holds  in  our  educational  system,  deserves  it  mainly  on  the 
ground  that  it  is  to  be  treated  as  a  logical  exercise.  It  is  the 
only  branch  of  mathematics  which  has  found  its  way  into 
primary  and  early  education;  other  departments  of  pure  science 
being  reserved  for  what  is  called  higher  or  university  instruction. 
But  all  the  arguments  in  favor  of  teaching  algebra  and  trig- 
onometry to  advanced  students,  apply  equally  to  the  teaching 
of  the  principles  or  theory  of  arithmetic  to  schoolboys.  It  is 
calculated  to  do  for  them  exactly  the  same  kind  of  service,  to 
educate  one  side  of  their  minds,  to  bring  into  play  one  set  of 
faculties  which  cannot  be  so  severely  or  properly  exercised  in  any 
other  department  of  learning.  In  short,  relatively  to  the  needs 
of  a  beginner,  Arithmetic,  as  a  science,  is  just  as  valuable — it 
is  certainly  quite  as  intelligible — as  the  higher  mathematics  to 
a  university  student. — FITCH,  J.  G. 

Lectures    on    Teaching    (New    York,    1906), 
pp.  267-268. 

1625.  What  mathematics,  therefore  are  expected  to  do  for  the 
advanced  student  at  the  university,  Arithmetic,  if  taught  de- 
monstratively, is  capable  of  doing  for  the  children  even  of  the 
humblest  school.  It  furnishes  training  in  reasoning,  and  partic- 
ularly in  deductive  reasoning.  It  is  a  discipline  in  closeness 
and  continuity  of  thought.  It  reveals  the  nature  of  fallacies, 
and  refuses  to  avail  itself  of  unverified  assumptions.  It  is  the 
one  department  of  school-study  in  which  the  sceptical  and  in- 
quisitive spirit  has  the  most  legitimate  scope;  in  which  authority 
goes  for  nothing.  In  other  departments  of  instruction  you  have 
a  right  to  ask  for  the  scholar's  confidence,  and  to  expect  many 
things  to  be  received  on  your  testimony  with  the  understand- 
ing that  they  will  be  explained  and  verified  afterwards.  But 
here  you  are  justified  in  saying  to  your  pupil  "Believe  nothing 
which  you  cannot  understand.  Take  nothing  for  granted."  In 
short,  the  proper  office  of  arithmetic  is  to  serve  as  elementary 


268  MEMORABILIA   MATHEMATICA 

training  in  logic.  All  through  your  work  as  teachers  you  will 
bear  in  mind  the  fundamental  difference  between  knowing  and 
thinking;  and  will  feel  how  much  more  important  relatively  to 
the  health  of  the  intellectual  life  the  habit  of  thinking  is  than 
the  power  of  knowing,  or  even  facility  of  achieving  visible  re- 
sults. But  here  this  principle  has  special  significance.  It  is  by 
Arithmetic  more  than  by  any  other  subject  in  the  school  course 
that  the  art  of  thinking — consecutively,  closely,  logically — can 
be  effectually  taught. — FITCH,  J.  G. 

Lectures   on    Teaching    (New    York,    1906), 

1626.  Arithmetic  and  geometry,  those  wings  on  which  the 
astronomer  soars  as  high  as  heaven. — BOYLE,  ROBERT. 

Usefulness  of  Mathematics  to  Natural  Philos- 
ophy; Works  (London,  1772),  Vol.  8,  p.  429. 

1627.  Arithmetical  symbols  are  written  diagrams  and  geo- 
metrical figures  are  graphic  formulas. — HILBERT,  D. 

Mathematical  Problems;  Bulletin  American 
Mathematical  Society,  Vol.  8  (1902),  p.  443. 

1628.  Arithmetic  and  geometry  are  much  more  certain  than 
the  other  sciences,  because  the  objects  of  them  are  in  themselves 
so  simple  and  so  clear  that  they  need  not  suppose  anything  which 
experience  can  call  in  question,  and  both  proceed  by  a  chain  of 
consequences  which  reason  deduces  one  from  another.    They 
are  also  the  easiest  and  clearest  of  all  the  sciences,  and  their 
object  is  such  as  we  desire;  for,  except  for  want  of  attention,  it  is 
hardly  supposable  that  a  man  should  go  astray  in  them.    We 
must  not  be  surprised,  however,  that  many  minds  apply  them- 
selves by  preference  to  other  studies,  or  to  philosophy.    Indeed 
everyone  allows  himself  more  freely  the  right  to  make  his  guess 
if  the  matter  be  dark  than  if  it  be  clear,  and  it  is  much  easier  to 
have  on  any  question  some  vague  ideas  than  to  arrive  at  the 
truth  itself  on  the  simplest  of  all. — DESCARTES. 

Rules  for  the  Direction  of  the  Mind;  Torrey's 
Philosophy  of  Descartes  (New  York,  1892), 
p.  63. 


ARITHMETIC  269 

1629.  Why  are  wise  few,  fools  numerous  in  the  excesse? 
'Cause,  wanting  number,  they  are  numberlesse. 

LOVELACE. 

Noah  Bridges:  Vulgar  Arithmetike  (London, 
1659),  p.  127. 

1630.  The  clearness  and  distinctness  of  each  mode  of  number 
from  all  others,  even  those  that  approach  nearest,  makes  me 
apt  to  think  that  demonstrations  hi  numbers,  if  they  are  not 
more  evident  and  exact  than  in  extension,  yet  they  are  more 
general  hi  their  use,  and  more  determinate  in  their  application. 
Because  the  ideas  of  numbers  are  more  precise  and  distinguish- 
able than  in  extension;  where  every  equality  and  excess  are  not 
so  easy  to  be  observed  or  measured;  because  our  thoughts  can- 
not in  space  arrive  at  any  determined  smallness  beyond  which 
it  cannot  go,  as  an  unit;  and  therefore  the  quantity  or  proportion 
of  any  the  least  excess  cannot  be  discovered. — LOCKE,  JOHN. 

An  Essay  concerning  Human  Understanding, 
Bk.  2,  chap.  16,  sect.  4- 

1631.  Battalions  of  figures  are  like  battalions  of  men,  not 
always  as  strong  as  is  supposed. — SAGE,  M. 

Mrs.  Piper  and  the  Society  for  Psychical  Re- 
search [Robertson]  (New  York,  1909),  p.  151. 

1632.  Number  was  born  in  superstition  and  reared  in  mys- 
tery, .  .  .  numbers  were  once  made  the  foundation  of  religion 
and  philosophy,  and  the  tricks  of  figures  have  had  a  marvellous 
effect  on  a  credulous  people. — PARKER,  F.  W. 

Talks  on  Pedagogics  (New  York,  1894),  P-  64- 

1633.  A  rule  to  trick  th'  arithmetic. — KIPLING,  R. 

To  the  True  Romance. 

1634.  God  made  integers,  all  else  is  the  work  of  man. 

KRONECKER,  L. 

Jahresberichte    der    Deutschen    Mathematiker 
Vereinigung,  Bd.  2,  p.  19. 

1635.  Plato   said   "aet  6  tfeo?  ycwfjieTpei."     Jacobi  changed 
this   to   "ael   6   0eo9    aptO/jujri&t."     Then   came   Kronecker 


270  MEMORABILIA    MATHEMATICA 

and  created  the  memorable  expression  "Die  ganzen  Zahlen  hat 
Gott  gemacht,  alles  andere  ist  Menschenwerk." — KLEIN,  F. 

Jahresbericht    der    Deutschen    Mathematiker 

Vereinigung,  Bd.  6,  p.  186. 

1636.  Integral  numbers  are  the  fountainhead  of  all  mathe- 
matics.— MINKOWSKI,  H. 

Diophantische    Approximationen     (Leipzig, 
1907),  Vorrede. 

1637.  The  "  Disquisitiones  Arithmeticae"  that  great  book 
with  seven  seals. — MERZ,  J.  T. 

A  History  of  European  Thought  in  the  Nine- 
teenth Century  (Edinburgh  and  London,  1903), 
p.  721. 

1638.  It  may  fairly  be  said  that  the  germs  of  the  modern 
algebra  of  linear  substitutions  and  concomitants  are  to  be 
found  in  the  fifth  section  of  the  Disquisitiones  Arithmeticae;  and 
inversely,  every  advance  in  the  algebraic  theory  of  forms  is  an 
acquisition  to  the  arithmetical  theory. — MATHEWS,  G.  B. 

Theory  of  Numbers  (Cambridge,  1892),  Part  1, 
sect.  48. 

1639.  Strictly  speaking,  the  theory  of  numbers  has  nothing  to 
do  with  negative,  or  fractional,  or  irrational  quantities,  as  such. 
No  theorem  which  cannot  be  expressed  without  reference  to 
these  notions  is  purely  arithmetical:  and  no  proof  of  an  arith- 
metical theorem,  can  be  considered  finally  satisfactory  if  it 
intrinsically  depends  upon  extraneous  analytical  theories. 

MATHEWS,  G.  B. 

Theory  of  Numbers  (Cambridge,  1892),  Part  1, 
sect.  1. 

1640.  Many  of  the  greatest  masters  of  the  mathematical 
sciences  were  first  attracted  to  mathematical  inquiry  by  prob- 
lems relating  to  numbers,  and  no  one  can  glance  at  the  periodi- 
cals of  the  present  day  which  contain  questions  for  solution 
without  noticing  how  singular  a  charm  such  problems  still 
continue  to  exert.    The  interest  in  numbers  seems  implanted 
in  the  human  mind,  and  it  is  a  pity  that  it  should  not  have 
freer  scope  in  this  country.    The  methods  of  the  theory  of  num- 


ARITHMETIC  271 

bers  are  peculiar  to  itself,  and  are  not  readily  acquired  by  a 
student  whose  mind  has  for  years  been  familiarized  with  the 
very  different  treatment  which  is  appropriate  to  the  theory  of 
continuous  magnitude;  it  is  therefore  extremely  desirable  that 
some  portion  of  the  theory  should  be  included  hi  the  ordinary 
course  of  mathematical  instruction  at  our  University.  From  the 
moment  that  Gauss,  in  his  wonderful  treatise  of  1801,  laid  down 
the  true  lines  of  the  theory,  it  entered  upon  a  new  day,  and 
no  one  is  likely  to  be  able  to  do  useful  work  in  any  part  of  the 
subject  who  is  unacquainted  with  the  principles  and  concep- 
tions with  which  he  endowed  it. — GLAISHER,  J.  W.  L. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science  (1890};  Nature, 
Vol.  42,  p.  467. 

1641.  Let  us  look  for  a  moment  at  the  general  significance  of 
the  fact  that  calculating  machines  actually  exist,  which  relieve 
mathematicians  of  the  purely  mechanical  part  of  numerical 
computations,  and  which  accomplish  the  work  more  quickly  and 
with  a  greater  degree  of  accuracy;  for  the  machine  is  not  sub- 
ject to  the  slips  of  the  human  calculator.     The  existence  of 
such  a  machine  proves  that  computation  is  not  concerned  with 
the  significance  of  numbers,  but  that  it  is  concerned  essentially 
only  with  the  formal  laws  of  operation;  for  it  is  only  these  that 
the  machine   can   obey — having  been  thus   constructed — an 
intuitive  perception  of  the  significance  of  numbers  being  out  of 
the  question. — KLEIN,  F. 

Elementarmathematik  vom  hoheren  Stand- 
punkte  aus.  (Leipzig,  1908),  Bd.  1,  p.  58. 

1642.  Mathematics  is  the  queen  of  the  sciences  and  arith- 
metic the  queen  of  mathematics.     She  often  condescends  to 
render  service  to  astronomy  and  other  natural  sciences,  but  in 
all  relations  she  is  entitled  to  the  first  rank. — GAUSS. 

Sartorius  von  W  alter  shausen:  Gauss  zum 
Gedachtniss.  (Leipzig,  1856),  p.  79. 

1643.  Zu  Archimedes  kam  ein  wissbegieriger  Jiingling, 
Weihe  mich,  sprach  er  zu  ihm,  ein  hi  die  gottliche 

Kunst, 


272  MEMORABILIA    MATHEMATICA 

Die  so  herrliche  Dienste  der  Sternenkunde  geleistet, 
Hinter  dem  Uranos  noch  einen  Planeten  entdeckt. 
Gottlieb  nennst  Du  die  Kunst,  sie  ist's,  versetzte  der 

Weise, 

Aber  sie  war  es,  bevor  noch  sie  den  Kosmos  erforscht, 
Ehe  sie  herrliche  Dienste  der  Sternenkunde  geleistet, 
Hinter  dem  Uranos  noch  einen  Planeten  entdeckt. 
Was  Du  un  Kosmos  erblickst,  ist  nur  der  Gottlichen 

Abglanz, 
In  der  Olympier  Schaar  thronet  die  ewige  Zahl. 

JACOBI,  C.  G.  J. 

Journal  fur  Mathematik,  Bd.  101  (1887),  p. 
338. 

To  Archimedes  came  a  youth  intent  upon  knowledge, 

Quoth  he,  "Initiate  me  into  the  science  divine 

Which  to  astronomy,  lo!  such  excellent  service  has 
rendered, 

And  beyond  Uranus'  orb  a  hidden  planet  revealed." 

"Call'st  thou  the  science  divine?  So  it  is,"  the  wise 
man  responded, 

"But  so  it  was  long  before  its  light  on  the  Cosmos  it 
shed, 

Ere  hi  astronomy's  realm  such  excellent  service  it 
rendered, 

And  beyond  Uranus'  orb  a  hidden  planet  revealed. 

Only  reflection  divine  is  that  which  Cosmos  dis- 
closes, 

Number  herself  sits  enthroned  among  Olympia's  hosts. 

1644.  The  higher  arithmetic  presents  us  with  an  inexhaustible 
store  of  interesting  truths, — of  truths  too,  which  are  not  isolated, 
but  stand  in  a  close  internal  connexion,  and  between  which,  as 
our  knowledge  increases,  we  are  continually  discovering  new 
and  sometimes  wholly  unexpected  ties.  A  great  part  of  its 
theories  derives  an  additional  charm  from  the  peculiarity  that 
important  propositions,  with  the  impress  of  simplicity  upon 
them,  are  often  easily  discoverable  by  induction,  and  yet  are  of 
so  profound  a  character  that  we  cannot  find  their  demonstration 


ARITHMETIC  273 

till  after  many  vain  attempts;  and  even  then,  when  we  do 
succeed,  it  is  often  by  some  tedious  and  artificial  process,  while 
the  simpler  methods  may  long  remain  concealed. 

GAUSS,  C.  F. 

Preface  to  Eisenstein's  Mathematische  Ab- 
handlungen  (Berlin,  1847),  [H.  J.  S.  Smith]. 

1645.  The  Theory  of  Numbers  has  acquired  a  great  and  in- 
creasing claim  to  the  attention  of  mathematicians.    It  is  equally 
remarkable  for  the  number  and  importance  of  its  results,  for 
the  precision  and  rigorousness  of  its  demonstrations,  for  the 
variety  of  its  methods,  for  the  intimate  relations  between  truths 
apparently  isolated  which  it  sometimes  discloses,  and  for  the 
numerous  applications  of  which  it  is  susceptible  in  other  parts  of 
analysis. — SMITH,  H.  J.  S. 

Report  on  the  Theory  of  Numbers,  British 
Association,  1859;  Collected  Mathematical 
Papers,  Vol.  1,  p.  88. 

1646.  The  invention  of  the  symbol  =  by  Gauss  affords  a 
striking  example  of  the  advantage  which  may  be  derived  from 
an  appropriate  notation,  and  marks  an  epoch  in  the  development 
of  the  science  of  arithmetic. — MATHEWS,  G.  B. 

Theory  of  Numbers  (Cambridge,  1892),  Part  1, 
sect.  29. 

1647.  As  Gauss  first  pointed  out,  the  problem  of  cyclotomy, 
or  division  of  the  circle  into  a  number  of  equal  parts,  depends 
in  a  very  remarkable  way  upon  arithmetical  considerations. 
We  have  here  the  earliest  and  simplest  example  of  those  rela- 
tions of  the  theory  of  numbers  to  transcendental  analysis,  and 
even  to  pure  geometry,  which  so  often  unexpectedly  present 
themselves,  and  which,  at  first  sight,  are  so  mysterious. 

MATHEWS,  G.  B. 

Theory  of  Numbers  (Cambridge,  1892),  Part  1, 
sect.  167. 

1648.  I  have  sometimes  thought  that  the  profound  mystery 
which  envelops  our  conceptions  relative  to  prime  numbers  de- 
pends upon  the  limitations  of  our  faculties  in  regard  to  time, 


274  MEMORABILIA   MATHEMATICA 

which  like  space  may  be  in  its  essence  poly-dimensional,  and 
that  this  and  such  sort  of  truths  would  become  self-evident  to  a 
being  whose  mode  of  perception  is  according  to  superficially 
as  distinguished  from  our  own  limitation  to  linearly  extended 
time. — SYLVESTER,  J.  J. 

Collected  Mathematical  Papers,  Vol.  4,  P-  600, 

footnote. 


CHAPTER  XVII 

ALGEBRA 

1701.  The  science  of  algebra,  independently  of  any  of  its 
uses,  has  all  the  advantages  which  belong  to  mathematics  in 
general  as  an  object  of  study,  and  which  it  is  not  necessary  to 
enumerate.     Viewed  either  as  a  science  of  quantity,  or  as  a 
language  of  symbols,  it  may  be  made  of  the  greatest  service  to 
those  who  are  sufficiently  acquainted  with  arithmetic,  and  who 
have  sufficient  power  of  comprehension  to  enter  fairly  upon  its 
difficulties. — DE  MORGAN,  A. 

Elements  of  Algebra  (London,  1887),  Preface. 

1702.  Algebra  is  generous,  she  often  gives  more  than  is  asked 
of  her. — D  'ALEMBERT. 

Quoted   in   Bulletin  American   Mathematical 
Society,  Vol.  2  (1905),  p.  285. 

1703.  The  operations  of  symbolic  arithmetick  seem  to  me  to 
afford  men  one  of  the  clearest  exercises  of  reason  that  I  ever 
yet  met  with,  nothing  being  there  to  be  performed  without 
strict  and  watchful  ratiocination,  and  the  whole  method  and 
progress  of  that  appearing  at  once  upon  the  paper,  when  the 
operation  is  finished,  and  affording  the  analyst  a  lasting,  and, 
as  it  were,  visible  ratiocination. — BOYLE,  ROBERT. 

Works  (London,  1772),  Vol.  3,  p.  426. 

1704.  The  human  mind  has  never  invented  a  labor-saving 
machine  equal  to  algebra. — 

The  Nation,  Vol.  S3,  p.  237. 

1705.  They  that  are  ignorant  of  Algebra  cannot  imagine  the 
wonders  in  this  kind  are  to  be  done  by  it:  and  what  further 
improvements  and  helps  advantageous  to  other  parts  of  knowl- 
edge the  sagacious  mind  of  man  may  yet  find  out,  it  is  not  easy 
to  determine.    This  at  least  I  believe,  that  the  ideas  of  quantity 

275 


276  MEMORABILIA   MATHEMATICA 

are  not  those  alone  that  are  capable  of  demonstration  and 
knowledge;  and  that  other,  and  perhaps  more  useful,  parts  of 
contemplation,  would  afford  us  certainty,  if  vices,  passions,  and 
domineering  interest  did  not  oppose  and  menace  such  en- 
deavours.— LOCKE,  JOHN. 

An  Essay  concerning  Human  Understanding, 

Bk.  4,  chap.  3,  sect.  18. 

1706.  Algebra  is  but  written  geometry  and  geometry  is  but 
figured  algebra. — GERMAIN,  SOPHIE. 

Memoire  sur  la  surfaces  elastiques. 

1707.  So  long  as  algebra  and  geometry  proceeded  separately 
their  progress  was  slow  and  their  application  limited,  but  when 
these  two  sciences  were  united,  they  mutually  strengthened  each 
other,  and  marched  together  at  a  rapid  pace  toward  perfection. 

LAGRANGE. 

Leqons  eUmentaires  sur  les  Mathematiques, 
Leqon  Cinquieme. 

1708.  The  laws  of  algebra,  though  suggested  by  arithmetic, 
do  not  depend  on  it.    They  depend  entirely  on  the  conventions 
by  which  it  is  stated  that  certain  modes  of  grouping  the  symbols 
are  to  be  considered  as  identical.    This  assigns  certain  prop- 
erties to  the  marks  which  form  the  symbols  of  algebra.    The 
laws  regulating  the  manipulation  of  algebraic  symbols  are 
identical  with  those  of  arithmetic.    It  follows  that  no  algebraic 
theorem  can  ever  contradict  any  result  which  could  be  arrived  at 
by  arithmetic;  for  the  reasoning  in  both  cases  merely  applies 
the  same  general  laws  to  different  classes  of  things.     If  an 
algebraic  theorem  can  be  interpreted  in  arithmetic,  the  cor- 
responding arithmetical  theorem  is  therefore  true. 

WHITEHEAD,  A.  N. 
Universal  Algebra  (Cambridge,  1898),  p.  2. 

1709.  That  a  formal  science  like  algebra,  the  creation  of  our 
abstract  thought,  should  thus,  in  a  sense,  dictate  the  laws  of  its 
own  being,  is  very  remarkable.    It  has  required  the  experience 
of  centuries  for  us  to  realize  the  full  force  of  this  appeal. 

MATHEWS,  G.  B. 

F.  Spencer:  Chapters  on  Aims  and  Practice  of 
Teaching  (London,  1899),  p.  184. 


ALGEBRA  277 

1710.  The  rules  of  algebra  may  be  investigated  by  its  own 
principles,  without  any  aid  from  geometry;  and  although  in 
many  cases  the  two  sciences  may  serve  to  illustrate  each  other, 
there  is  not  now  the  least  necessity  in  the  more  elementary 
parts  to  call  in  the  aid  of  the  latter  in  expounding  the  former. 

CHRYSTAL,  GEORGE. 

Encyclopedia  Britannica,  9th  Edition;  Article 
"Algebra." 

1711.  Algebra,  as  an  art,  can  be  of  no  use  to  any  one  in  the 
business  of  life;  certainly  not  as  taught  in  the  schools.    I  appeal 
to  every  man  who  has  been  through  the  school  routine  whether 
this  be  not  the  case.    Taught  as  an  art  it  is  of  little  use  in  the 
higher  mathematics,  as  those  are  made  to  feel  who  attempt  to 
study  the  differential  calculus  without  knowing  more  of  the 
principles  than  is  contained  in  books  of  rules. 

DE  MORGAN,  A. 
Elements  of  Algebra  (London,  1837),  Preface. 

1712.  We  may  always  depend  upon  it  that  algebra,  which 
cannot  be  translated  into  good  English  and  sound  common 
sense,  is  bad  algebra  — CLIFFORD,  W.  K. 

Common  Sense  in  the  Exact  Sciences  (London, 
1885),  chap.  1,  sect.  7. 

1713.  The  best  review  of  arithmetic  consists  in  the  study  of 
algebra. — CAJORI,  F. 

Teaching  and  History  of  Mathematics  in  U.  S. 
(Washington,  1896),  p.  110. 

1714.  [Algebra]  has  for  its  object  the  resolution  of  equations; 
taking  this  expression  in  its  full  logical  meaning,  which  signifies 
the  transformation  of  implicit  functions  into  equivalent  explicit 
ones.    In  the  same  way  arithmetic  may  be  defined  as  destined  to 
the  determination  of  the  values  of  functions.  .  .  .  We  will 
briefly  say  that  Algebra  is  the  Calculus  of  Functions,  and  Arith- 
metic the  Calculus  of  Values. — COMTE,  A. 

Philosophy  of  Mathematics  [Gillespie]  (New 
York,  1851),  p.  55. 


278  MEMORABILIA    MATHEMATICA 

1716.  .  .  .  the  subject  matter  of  algebraic  science  is  the 
abstract  notion  of  time;  divested  of,  or  not  yet  clothed  with,  any 
actual  knowledge  which  we  may  possess  of  the  real  Events  of 
History,  or  any  conception  which  we  may  frame  of  Cause  and 
Effect  in  Nature;  but  involving,  what  indeed  it  cannot  be  di- 
vested of,  the  thought  of  possible  Succession,  or  of  pure,  ideal 
Progression. — HAMILTON,  W.  R. 

Graves'  Life  of  Hamilton  (New  York,  1882- 
1889),  Vol.  3,  p.  633. 

1716.  .  .  .  instead   of   seeking   to   attain   consistency   and 
uniformity  of  system,  as  some  modern  writers  have  attempted, 
by  banishing  this  thought  of  time  from  the  higher  Algebra,  I 
seek  to  attain  the  same  object,  by  systematically  introducing  it 
into  the  lower  or  earlier  parts  of  the  science  — HAMILTON,  W.  R. 

Graves'  Life  of  Hamilton  (New  York,  1882- 
1889),  Vol.  3,  p.  684. 

1717.  The  circumstances  that  algebra  has  its  origin  in  arith- 
metic, however  widely  it  may  in  the  end  differ-  from   that 
science,  led  Sir  Isaac  Newton  to  designate  it  "Universal  Arith- 
metic," a  designation  which,  vague  as  it  is,  indicates  its  charac- 
ter better  than  any  other  by  which  it  has  been  attempted  to 
express  its  functions — better  certainly,  to  ordinary  minds,  than 
the  designation  which  has  been  applied  to  it  by  Sir  William 
Rowan  Hamilton,  one  of  the  greatest  mathematicians  the  world 
has  seen  since  the  days  of  Newton — "the  Science  of  Pure 
Time;"  or  even  than  the  title  by  which  De  Morgan  would 
paraphrase  Hamilton's  words — "the  Calculus  of  Succession." 

CHBYSTAL,  GEOR  E. 

Encyclopedia  Britannica,  9th  Edition;  Article 
"Algebra." 

1718.  Time  is  said  to  have  only  one  dimension,  and  space  to 
have  three  dimensions.  .  .  .  The  mathematical  quaternion  par- 
takes of  both  these  elements;  in  technical  language  it  may  be 
said  to  be  "tune  plus  space,"  or  "space  plus  time:"  and  in  this 
sense  it  has,  or  at  least  involves  a  reference  to,  four  dimen- 
sions. . 


ALGEBRA  279 

And  how  the  One  of  Time,  of  Space  the  Three, 
Might  in  the  Chain  of  Symbols  girdled  be. 

HAMILTON,  W.  R. 

Graves'  Life  of  Hamilton  (New  York,  1882- 
1889),  Vol.  3,  p.  685. 

1719.  It  is  confidently  predicted,  by  those  best  qualified  to 
judge,  that  in  the  coming  centuries  Hamilton's  Quaternions  will 
stand  out  as  the  great  discovery  of  our  nineteenth  century. 
Yet  how  silently  has  the  book  taken  its  place  upon  the  shelves  of 
the  mathematician's  library!    Perhaps  not  fifty  men  on  this 
side  of  the  Atlantic  have  seen  it,  certainly  not  five  have  read  it. 

HILL,  THOMAS. 
North  American  Review,  Vol.  85,  p.  228. 

1720.  I  think  the  time  may  come  when  double  algebra  will 
be  the  beginner's  tool;  and  quaternions  will  be  where  double 
algebra  is  now.    The  Lord  only  knows  what  will  come  above  the 
quaternions. — DE  MORGAN,  A. 

Graves'  Life  of  Hamilton  (New  York,  1882- 
1889),  Vol.  3,  p.  498. 

1721.  Quaternions  came  from  Hamilton  after  his  really  good 
work  had  been  done;  and  though  beautifully  ingenious,  have 
been  an  unmixed  evil  to  those  who  have  touched  them  in  any 
way,  including  Clerk  Maxwell. — THOMSON,  WILLIAM. 

Thompson,  S.  P.:  Life  of  Lord  Kelvin  (London, 
1910),  p.  1138. 

1722.  The  whole  affair  [quaternions]  has  in  respect  to  mathe- 
matics a  value  not  inferior  to  that  of  "Volapuk"  in  respect  to 
language. — THOMSON,  WILLIAM. 

Thompson,  S.  P.:  Life  of  Lord  Kelvin  (London, 
1910),  p.  1188. 

1723.  A  quaternion  of  maladies!    Do  send  me  some  formula 
by  help  of  which  I  may  so  doctor  them  that  they  may  all  become 
imaginary  or  positively  equal  to  nothing. — SEDGWICK. 

Graves'  Life  of  Hamilton  (New  York,  1882- 
1889),  Vol.  3,  p.  2. 


280  MEMORABILIA   MATHEMATICA 

1724.  If  nothing  more  could  be  said  of  Quaternions  than  that 
they  enable  us  to  exhibit  in  a  singularly  compact  and  elegant 
form,  whose  meaning  is  obvious  at  a  glance  on  account  of  the 
utter  inartificiality  of  the  method,  results  which  in  the  ordinary 
Cartesian  co-ordinates  are  of  the  utmost  complexity,  a  very 
powerful  argument  for  their  use  would  be  furnished.    But  it 
would  be  unjust  to  Quaternions  to  be  content  with  such  a  state- 
ment; for  we  are  fully  entitled  to  say  that  in  all  cases,  even  in 
those  to  which  the  Cartesian  methods  seem  specially  adapted, 
they  give  as  simple  an  expression  as  any  other  method;  while  in 
the  great  majority  of  cases  they  give  a  vastly  simpler  one.    In 
the  common  methods  a  judicious  choice  of  co-ordinates  is  often 
of  immense   importance  in  simplifying  an  investigation;  in 
Quaternions  there  is  usually  no  choice,  for  (except  when  they 
degrade  to  mere  scalars)  they  are  in  general  utterly  independent 
of  any  particular  directions  hi  space,  and  select  of  themselves  the 
most  natural  reference  lines  for  each  particular  problem. 

TAIT,  P.  G 

Presidential  Address  British  Association  for 
the  Advancement  of  Science  (1871);  Nature, 
Vol.  4,  p.  270. 

1725.  Comparing  a  Quaternion  investigation,  no  matter  in 
what  department,  with  the  equivalent  Cartesian  one,  even 
when  the  latter  has  availed  itself  to  the  utmost  of  the  improve- 
ments suggested  by  Higher  Algebra,  one  can  hardly  help  making 
the  remark  that  they  contrast  even  more  strongly  than  the 
decimal  notation  with  the  binary  scale,  or  with  the  old  Greek 
arithmetic — or  than  the  well-ordered  subdivisions  of  the  metri- 
cal system  with  the  preposterous  no-systems  of  Great  Britain, 
a  mere  fragment  of  which  (hi  the  form  of  Table  of  Weights  and 
Measures)  form,  perhaps  the  most  effective,  if  not  the  most 
ingenious,  of  the  many  instruments  of  torture  employed  in  our 
elementary  teaching. — TAIT,  P.  G. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science  (1871);  Nature, 
Vol.  4,  P.  ^71. 

1726.  It  is  true  that,  in  the  eyes  of  the  pure  mathematician, 
Quaternions  have  one  grand  and  fatal  defect.    They  cannot  be 


ALGEBRA  281 

applied  to  space  of  n  dimensions,  they  are  contented  to  deal  with 
those  poor  three  dimensions  in  which  mere  mortals  are  doomed 
to  dwell,  but  which  cannot  bound  the  limitless  aspirations  of  a 
Cayley  or  a  Sylvester.  From  the  physical  point  of  view  this, 
instead  of  a  defect,  is  to  be  regarded  as  the  greatest  possible 
recommendation.  It  shows,  in  fact,  Quaternions  to  be  the 
special  instrument  so  constructed  for  application  to  the  Actual 
as  to  have  thrown  overboard  everything  which  is  not  absolutely 
necessary,  without  the  slightest  consideration  whether  or  no  it 
was  thereby  being  rendered  useless  for  application  to  the  In- 
conceivable.— TAIT,  P.  G. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science  (1871);  Nature, 
Vol.  4,  p.  271. 

1727.  There  is  an  old  epigram  which  assigns  the  empire  of  the 
sea  to  the  English,  of  the  land  to  the  French,  and  of  the  clouds 
to  the  Germans.    Surely  it  was  from  the  clouds  that  the  Ger- 
mans fetched  +  and  -  ;  the  ideas  which  these  symbols  have 
generated  are  much  too  important  to  the  welfare  of  humanity  to 
have  come  from  the  sea  or  from  the  land. — WHITEHEAD,  A.  N. 

An  Introduction  to  Mathematics  (New  York, 
1911),  p.  86. 

1728.  Now  as  to  what  pertains  to  these  Surd  numbers  (which, 
as  it  were  by  way  of  reproach  and  calumny,  having  no  merit  of 
their  own  are  also  styled  Irrational,  Irregular,  and  Inexplicable) 
they  are  by  many  denied  to  be  numbers  properly  speaking,  and 
are  wont  to  be  banished  from  arithmetic  to  another  Science, 
(which  yet  is  no  science)  viz.  algebra. — BARROW,  ISAAC. 

Mathematical  Lectures  (London,  1734),  p.  44- 

1729.  If  it  is  true  as  Whewell  says,  that  the  essence  of  the 
triumphs  of  science  and  its  progress  consists  in  that  it  enables  us 
to  consider  evident  and  necessary,  views  which  our  ancestors 
held  to  be  unintelligible  and  were  unable  to  comprehend,  then 
the  extension  of  the  number  concept  to  include  the  irrational, 
and  we  will  at  once  add,  the  imaginary,  is  the  greatest  forward 
step  which  pure  mathematics  has  ever  taken. 

HANKEL,  HERMANN. 

Theorie  der  Complexen  Zahlen  (Leipzig,  1867), 
p.  60. 


282  MEMORABILIA   MATHEMATICA 

1730.  That    this    subject    [of    imaginary    magnitudes]    has 
hitherto  been  considered  from  the  wrong  point  of  view  and 
surrounded  by  a  mysterious  obscurity,   is  to  be  attributed 
largely  to  an  ill-adapted  notation.     If  for  instance,   +1,  —  1, 
V—  1  had  been  called  direct,  inverse,  and  lateral  units,  instead 
of  positive,  negative,  and  imaginary  (or  even  impossible)  such 
an  obscurity  would  have  been  out  of  question. — GAUSS,  C.  F. 

Theoria  residiorum  biquadraticorum,  Commen- 
tatio  secunda;  Werke,  Bd.  2  (Goettingen,  1868), 
p.  177. 

1731.  .  .  .  the    imaginary,    this    bosom-child    of    complex 
mysticism. — DUHRING,  EUGEN. 

Kritische  Geschichte  der  allgemeinen  Principien 
der  Mechanik  (Leipzig,  1877),  p.  517. 

1732.  Judged  by  the  only  standards  which  are  admissible  in 
a  pure  doctrine  of  numbers  i  is  imaginary  in  the  same  sense  as 
the  negative,  the  fraction,  and  the  irrational,  but  in  no  other 
sense;  all  are  alike  mere  symbols  devised  for  the  sake  of  rep- 
resenting the  results  of  operations  even  when  these  results  are 
not  numbers  (positive  integers). — FINE,  H.  B. 

The  Number-System  of  Algebra  (Boston,  1890), 
p.  86. 

1733.  This  symbol  [V  -  1]  is  restricted  to  a  precise  significa- 
tion as  the  representative  of  perpendicularity  in  quaternions, 
and  this  wonderful  algebra  of  space  is  intimately  dependent 
upon  the  special  use  of  the  symbol  for  its  symmetry,  elegance, 
and  power.    The  immortal  author  of  quaternions  has  shown 
that  there  are  other  significations  which  may  attach  to  the  sym- 
bol in  other  cases.    But  the  strongest  use  of  the  symbol  is  to 
be  found  in  its  magical  power  of  doubling  the  actual  universe, 
and  placing  by  its  side  an  ideal  universe,  its  exact  counterpart, 
with  which  it  can  be  compared  and  contrasted,  and,  by  means  of 
curiously  connecting  fibres,  form  with  it  an  organic  whole, 
from  which  modern  analysis  has  developed   her  surpassing 
geometry. — PEIRCE,  BENJAMIN. 

On  the  Uses  and  Transformations  of  Linear 
Algebras;  American  Journal  of  Mathematics, 
Vol.  4  (1881),  p.  216. 


ALGEBRA  283 

1734.  The  conception  of  the  inconceivable  [imaginary],  this 
measurement  of  what  not  only  does  not,  but  cannot  exist,  is 
one  of  the  finest  achievements  of  the  human  intellect.    No  one 
can  deny  that  such  imaginings  are  indeed  imaginary.     But 
they  lead  to  results  grander  than  any  which  flow  from  the 
imagination  of  the  poet.    The  imaginary  calculus  is  one  of  the 
masterkeys  to  physical  science.     These  realms  of  the  incon- 
ceivable afford  in  many  places  our  only  mode  of  passage  to  the 
domains  of  positive  knowledge.     Light  itself  lay  in  darkness 
until  this  imaginary  calculus  threw  light  upon  light.    And  in  all 
modern  researches  into  electricity,  magnetism,  and  heat,  and 
other  subtile  physical  inquiries,  these  are  the  most  powerful 
instruments. — HILL,  THOMAS. 

North  American  Review,  Vol.  85,  p.  235. 

1735.  All  the  fruitful  uses  of  imaginaries,  in  Geometry,  are 
those  which  begin  and  end  with  real  quantities,  and  use  imag- 
inaries only  for  the  intermediate  steps.    Now  in  all  such  cases, 
we  have  a  real  spatial  interpretation  at  the  beginning  and  end 
of  our  argument,  where  alone  the  spatial  interpretation  is 
important;  in  the  intermediate  links,  we  are  dealing  in  purely 
algebraic  manner  with  purely  algebraic  quantities,  and  may 
perform  any  operations  which  are  algebraically  permissible. 
If  the  quantities  with  which  we  end  are  capable  of  spatial  inter- 
pretation, then,  and  only  then,  our  results  may  be  regarded  as 
geometrical.    To  use  geometrical  language,  in  any  other  case,  is 
only  a  convenient  help  to  the  imagination.    To  speak,  for  exam- 
ple, of  projective  properties  which  refer  to  the  circular  points, 
is  a  mere  memoria  technica  for  purely  algebraical  properties; 
the  circular  points  are  not  to  be  found  in  space,  but  only  in  the 
auxiliary  quantities  by  which  geometrical  equations  are  trans- 
formed.    That  no  contradictions  arise  from  the  geometrical 
interpretation  of  imaginaries  is  not  wonderful;  for  they  are 
interpreted  solely  by  the  rules  of  Algebra,  which  we  may  admit 
as  valid  in  their  interpretation  to  imaginaries.    The  perception 
of  space  being  wholly  absent,  Algebra  rules  supreme,  and  no 
inconsistency  can  arise. — RUSSELL,  BERTRAND. 

Foundations  of  Geometry  (Cambridge,  1897), 
p.  45. 


284  MEMORABILIA   MATHEMATICA 

1736.  Indeed,  if  one  understands  by  algebra  the  application  of 
arithmetic  operations  to  composite  magnitudes  of  all  kinds, 
whether  they  be  rational  or  irrational  number  or  space  magni- 
tudes, then  the  learned  Brahmins  of  Hindostan  are  the  true 
inventors  of  algebra. — HANKEL,  HERMANN. 

Geschichte  der  Mathematik  im  Altertum  und 
Mittelalter  (Leipzig,  1874),  P-  195. 

1737.  It  is  remarkable  to  what  extent  Indian  mathematics 
enters  into  the  science  of  our  time.    Both  the  form  and  the 
spirit  of  the  arithmetic  and  algebra  of  modern  times  are  es- 
sentially Indian  and  not  Grecian. — CAJORI,  F. 

History  of  Mathematics   (New   York,   1897), 
p.  100. 

1738.  There  are  many  questions  in  this  science  [algebra] 
which  learned  men  have  to  this  time  in  vain  attempted  to  solve; 
and  they  have  stated  some  of  these  questions  in  their  writings, 
to  prove  that  this  science  contains  difficulties,  to  silence  those 
who  pretend  they  find  nothing  in  it  above  their  ability,  to 
warn  mathematicians  against  undertaking  to  answer  every 
question  that  may  be  proposed,  and  to  excite  men  of  genius  to 
attempt  their  solution.    Of  these  I  have  selected  seven. 

1.  To  divide  10  into  two  parts,  such,  that  when  each  part  is 
added  to  its  square-root  and  the  sums  multiplied  together,  the 
product  is  equal  to  the  supposed  number. 

2.  What  square  is  that,  which  being  increased  or  diminished 
by  10,  the  sum  and  remainder  are  both  square  numbers? 

3.  A  person  said  he  owed  to  Zaid  10  all  but  the  square-root  of 
what  he  owed  to  Amir,  and  that  he  owed  Amir  5  all  but  the 
square-root  of  what  he  owed  Zaid. 

4.  To  divide  a  cube  number  into  two  cube  numbers. 

5.  To  divide  10  into  two  parts  such,  that  if  each  is  divided  by 
the  other,  and  the  two  quotients  are  added  together,  the  sum  is 
equal  to  one  of  the  parts. 

6.  There  are  three  square  numbers  in  continued  geometric 
proportion,   such,    that   the   sum   of   the   three   is   a   square 
number. 

7.  There  is  a  square,  such,  that  when  it  is  increased  and 


ALGEBRA  285 

diminished  by  its  root  and  2,  the  sum  and  the  difference  are 
squares. — KHTJLASAT-AL-HISAB  . 

Algebra;  quoted  in  Hutton:  A  Philosophical 
and  Mathematical  Dictionary  (London,  1815}, 
Vol.  1,  p.  70. 

1739.  The  solution  of  such  questions  as  these  [referring  to  the 
solution  of  cubic  equations]  depends  on  correct  judgment,  aided 

by  the  assistance  of  God. — BIJA  GANITA. 

Quoted  in  Hutton:  A  Philosophical  and  Mathe- 
matical Dictionary  (London,  1815),  Vol.  1, 
p.  65. 

1740.  For  what  is  the  theory  of  determinants?    It  is  an  alge- 
bra upon  algebra;  a  calculus  which  enables  us  to  combine  and 
foretell  the  results  of  algebraical  operations,  in  the  same  way  as 
algebra  itself  enables  us  to  dispense  with  the  performance  of  the 
special  operations  of  arithmetic.    All  analysis  must  ultimately 
clothe  itself  under  this  form. — SYLVESTER,  J.  J. 

Philosophical  Magazine,  Vol.  1,  (1851),  p.  SOO; 
Collected  Mathematical  Papers,  Vol.  1,  p.  2^7. 

1741.  Fuchs.  Fast  mocht'  ich  imn  moderne  Algebra  studieren. 
Meph.  Ich  wiinschte  nicht  euch  irre  zu  fiihren. 

Was  diese  Wissenschaft  betrifft, 

Es  ist  so  schwer,  die  leere  Form  zu  meiden, 

Und  wenn  ihr  es  nicht  recht  begrifft, 

Vermogt    die    Indices    ihr    kaum    zu    unter- 

scheiden. 

Am  Besten  ist's,  wenn  ihr  nur  Einem  traut 
Und  auf  des  Meister's  Formeln  baut. 
Im  Ganzen — haltet  euch  an  die  Symbole. 
Dann  geht  ihr  zu  der  Forschung  Wohle 
Ins  sichre  Reich  der  Formeln  ein. 

Fuchs.  Ein  Resultat  muss  beim  Symbole  sein? 

Meph.  Schon  gut!  Nur  muss  man  sich  nicht  alzu 

angstlich  qualen. 

Denn  eben,  wo  die  Resultate  fehlen, 
Stellt  ein  Symbol  zur  rechten  Zeit  sich  ein. 
Symbolisch  lasst  sich  alles  schreiben, 
Miisst  nur  im  Allgemeinen  bleiben. 


286  MEMORABILIA   MATHEMATICA 

Wenn  man  der  Gleichung  Losung  nicht  er- 

kannte, 

Schreibt  man  sie  als  Determinante. 
Schreib '  was  du  willst,  nur  rechne  nie  was  aus. 
Symbole  lassen  trefflich  sich  traktieren, 
Mit  einem  Strich  1st  alles  auszufiihren, 
Und  mit  Symbolen  kommt  man  immer  aus. 

LASSWITZ,  KURD. 

Der  Faust-Tragodie  (-rc)ter  Teil;  Zeitschrift  fur 
mathematischen  und  naturwissenschaftlichen 
Unterricht,  Bd.  14,  p.  817. 

Fuchs.  To  study  modern  algebra  I'm  most  persuaded. 

Meph.  'Twas  not  my  wish  to  lead  thee  astray. 
But  as  concerns  this  science,  truly 
'Tis  difficult  to  avoid  the  empty  form, 
And  should'st  thou  lack  clear  comprehension, 
Scarcely  the  indices  thou '11  know  apart. 
'Tis  safest  far  to  trust  but  one 
And  built  upon  your  master's  formulas. 
On  the  whole — cling  closely  to  your  symbols. 
Then,  for  the  weal  of  research  you  may  gain 
An  entrance  to  the  formula's  sure  domain. 

Fuchs.  The  symbol,  it  must  lead  to  some  result? 

Meph.  Granted.    But  never  worry  about  results, 

For,   mind   you,   just  where  the   results  are 

wanting 

A  symbol  at  the  nick  of  time  appears. 
To  symbolic  treatment  all  things  yield, 
Provided  we  stay  in  the  general  field. 
Should  a  solution  prove  elusive, 
Write  the  equation  in  determinant  form. 
Write  what  you  please,  but  never  calculate. 
Symbols  are  patient  and  long  suffering, 
A  single  stroke  completes  the  whole  affair. 
Symbols  for  every  purpose  do  suffice. 

1742.  As  all  roads  are  said  to  lead  to  Rome,  so  I  find,  in  my 
own  case  at  least,  that  all  algebraic  inquiries  sooner  or  later  end 


ALGEBRA  287 

at  the  Capitol  of  Modern  Algebra  over  whose  shining  portal  is 
inscribed  "Theory  of  Invariants." — SYLVESTER,  J.  J. 

On  Newton's  Rule  for  the  Discovery  of  Imagi- 
nary Roots;  Collected  Mathematical  Papers, 
Vol.  2,  p.  380. 

1743.  If  we  consider  the  beauty  of  the  theorem  [Sylvester's 
Theorem  on  Newton's  Rule  for  the  Discovery  of  Imaginary 
Roots]  which  has  now  been  expounded,  the  interest  which  be- 
longs to  the  rule  associated  with  the  great  name  of  Newton, 
and  the  long  lapse  of  years  during  which  the  reason  and  extent 
of  that  rule  remained  undiscovered  by  mathematicians,  among 
whom  Maclaurin,  Waring  and  Euler  are  explicitly  included,  we 
must  regard  Professor  Sylvester's  investigations  made  to  the 
Theory  of  Equations  in  modern  times,  justly  to  be  ranked  with 
those  of  Fourier,  Sturm  and  Cauchy. — TODHUNTER,  I. 

Theory  of  Equations  (London,  1904),  P-  ®50. 

1744.  Considering  the  remarkable  elegance,  generality,  and 
simplicity  of  the  method   [Horner's  Method  of  finding  the 
numerical  values  of  the  roots  of  an  equation],  it  is  not  a  little 
surprising  that  it  has  not  taken  a  more  prominent  place  in 
current  mathematical  textbooks.  ...  As  a  matter  of  fact,  its 
spirit  is  purely  arithmetical;  and  its  beauty,  which  can  only  be 
appreciated  after  one  has  used  it  in  particular  cases,  is  of  that 
indescribably  simple  kind,  which  distinguishes  the  use  of  position 
in  the  decimal  notation  and  the  arrangement  of  the  simple 
rules  of  arithmetic.    It  is,  in  short,  one  of  those  things  whose 
invention  was  the  creation  of  a  commonplace. 

CHRYSTAL,  GEORGE. 

Algebra  (London  and  Edinburgh,  1893),  Vol.  1, 
chap.  15,  sect.  25. 

1745.  To  a  missing  member  of  a  family  group  of  terms  in  an 
algebraical  formula. 

Lone  and  discarded  one!  divorced  by  fate, 
Far  from  thy  wished-for  fellows — whither  art  flown? 
Where  lingerest  thou  in  thy  bereaved  estate, 
Like  some  lost  star,  or  buried  meteor  stone? 


288  MEMORABILIA    MATHEMATICA 

Thou  mindst  me  much  of  that  presumptuous  one 

Who  loth,  aught  less  than  greatest,  to  be  great, 

From  Heaven's  immensity  fell  headlong  down 

To  live  forlorn,  self-centred,  desolate: 

Or  who,  like  Heraclid,  hard  exile  bore, 

Now  buoyed  by  hope,  now  stretched  on  rack  of  fear, 

Till  throned  Astaea,  wafting  to  his  ear 

Words  of  dim  portent  through  the  Atlantic  roar, 

Bade  him  "the  sanctuary  of  the  Muse  revere 

And  strew  with  flame  the  dust  of  Isis'  shore." 

SYLVESTER,  J.  J. 

Inaugural    Lecture,    Oxford,    1885;    Nature, 

Vol.  S3,  p.  228. 

1746.  In  every  subject  of  inquiry  there  are  certain  entities, 
the  mutual  relations  of  which,  under  various  conditions,  it  is 
desirable  to  ascertain.  A  certain  combination  of  these  entities 
are  submitted  to  certain  processes  or  are  made  the  subjects  of 
certain  operations.  The  theory  of  invariants  in  its  widest 
scientific  meaning  determines  these  combinations,  elucidates 
their  properties,  and  expresses  results  when  possible  in  terms  of 
them.  Many  of  the  general  principles  of  political  science  and 
economics  can  be  represented  by  means  of  invariantive  rela- 
tions connecting  the  factors  which  enter  as  entities  into  the 
special  problems.  The  great  principle  of  chemical  science  which 
asserts  that  when  elementary  or  compound  bodies  combine  with 
one  another  the  total  weight  of  the  materials  is  unchanged,  is 
another  case  in  point.  Again,  in  physics,  a  given  mass  of  gas 
under  the  operation  of  varying  pressure  and  temperature  has 
the  well-known  invariant,  pressure  multiplied  by  volume  and 
divided  by  absolute  temperature.  ...  In  mathematics  the 
entities  under  examination  may  be  arithmetical,  algebraical, 
or  geometrical;  the  processes  to  which  they  are  subjected  may 
be  any  of  those  which  are  met  with  in  mathematical  work.  .  .  . 
It  is  the  principle  which  is  so  valuable.  It  is  the  idea  of  in- 
variance  that  pervades  today  all  branches  of  mathematics. 

MACMAHON,  P.  A. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science  (1901};  Nature, 
Vol.  64,  p.  481. 


ALGEBRA  289 

1747.  [The  theory  of  invariants]  has  invaded  the  domain  of 
geometry,  and  has  almost  re-created  the  analytical  theory;  but 
it  has  done  more  than  this  for  the  investigations  of  Cayley 
have  required  a  full  reconsideration  of  the  very  foundations  of 
geometry.     It  has  exercised  a  profound  influence  upon  the 
theory  of  algebraic  equations;  it  has  made  its  way  into  the 
theory  of  differential  equations;  and  the  generalisation  of  its 
ideas  is  opening  out  new  regions  of  most  advanced  and  profound 
functional  analysis.    And  so  far  from  its  course  being  completed, 
its  questions  fully  answered,  or  its  interest  extinct,  there  is  no 
reason  to  suppose  that  a  term  can  be  assigned  to  its  growth  and 

its  influence. — FORSYTH,  A.  R. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science  (1897);  Nature, 
Vol.  56,  p.  878. 

1748.  .  .  .  the  doctrine  of  Invariants,  a  theory  filling  the 
heavens  like  a  light-bearing  ether,  penetrating  all  the  branches 
of  geometry  and  analysis,  revealing  everywhere  abiding  con- 
figurations in  the  midst  of  change,  everywhere  disclosing  the 
eternal  reign  of  the  law  of  form. — KEYSER,  C.  J. 

Lectures  on  Science,  Philosophy  and  Art  (New 
York,  1908),  p.  28. 

1749.  It  is  in  the  mathematical  doctrine  of  Invariance,  the 
realm  wherein  are  sought  and  found  configurations  and  types  of 
being  that,  amidst  the  swirl  and  stress  of  countless  hosts  of 
transformations  remain  immutable,  and  the  spirit  dwells  in 
contemplation  of  the  serene  and  eternal  reign  of  the  subtile 
laws  of  Form,  it  is  there  that  Theology  may  find,  if  she  will,  the 
clearest  conceptions,  the  noblest  symbols,  the  most  inspiring 
intimations,  the  most  illuminating  illustrations,  and  the  surest 
guarantees  of  the  object  of  her  teaching  and  her  quest,  an 
Eternal  Being,  unchanging  in  the  midst  of  the  universal  flux. 

KEYSER,  C.  J. 

Lectures  on  Science,  Philosophy  and  Art  (New 
York,  1908),  p.  42. 

1750.  I  think  that  young  chemists  desirous  of  raising  their 
science  to  its  proper  rank  would  act  wisely  in  making  themselves 
master  betimes  of  the  theory  of  algebraic  forms.    What  mechan- 


290  MEMORABILIA   MATHEMATICA 

ics  is  to  physics,  that  I  think  is  algebraic  morphology,  founded 
at  option  on  the  theory  of  partitions  or  ideal  elements,  or  both, 
is  destined  to  be  to  the  chemistry  of  the  future.  .  .  .  invariants 
and  isomerism  are  sister  theories. — SYLVESTER,  J.  J. 

American  Journal  of  Mathematics,    Vol.    1 

(1878),  p.  126. 

1751.  The  great  notion  of  Group,  .  .  .  though  it  had  barely 
merged  into  consciousness  a  hundred  years  ago,  has  meanwhile 
become  a  concept  of  fundamental  importance  and  prodigious 
fertility,  not  only  affording  the  basis  of  an  imposing  doctrine — 
the  Theory  of  Groups — but  therewith  serving  also  as  a  bond  of 
union,  a  kind  of  connective  tissue,  or  rather  as  an  immense 
cerebro-spinal  system,  uniting  together  a  large  number  of  widely 
dissimilar  doctrines  as  organs  of  a  single  body. — KEYSER,  C.  J. 

Lectures  on  Science,  Philosophy  and  Art  (New 
York,  1908),  p.  12. 

1752.  In  recent  times  the  view  becomes  more  and  more 
prevalent  that  many  branches  of  mathematics  are  nothing  but 
the  theory  of  invariants  of  special  groups. — LIE,  SOPHUS. 

Continuierliche  Gruppen — Scheffers  (Leipzig, 
1898),  p.  665. 

1753.  Universal  Algebra  has  been  looked  on  with   some 
suspicion  by  many  mathematicians,  as  being  without  intrinsic 
mathematical  interest  and  as  being  comparatively  useless  as  an 
engine  of  investigation.  .  .  But  it  may  be  shown  that  Universal 
Algebra  has  the  same  claim  to  be  a  serious  subject  of  mathe- 
matical study  as  any  other  branch  of  mathematics. 

WHITEHEAD,  A.  N. 

Universal  Algebra  (Cambridge,  1898),  Preface, 
p.  vi. 

1754.  [Function]  theory  was,  in  effect,  founded  by  Cauchy; 
but,  outside  his  own  investigations,  it  at  first  made  slow  and 
hesitating  progress.    At  the  present  day,  its  fundamental  ideas 
may  be  said  almost  to  govern  most  departments  of  the  analysis 
of  continuous  quantity.    On  many  of  them,  it  has  shed  a  com- 
pletely new  light;  it  has  educed  relations  between  them  before 
unknown.    It  may  be  doubted  whether  any  subject  is  at  the 


ALGEBRA  291 

present  day  so  richly  endowed  with  variety  of  method  and 
fertility  of  resource;  its  activity  is  prodigious,  and  no  less 
remarkable  than  its  activity  is  its  freshness. — FORSYTH,  A.  R. 
Presidential  Address  British  Association  for 
the  Advancement  of  Science  (1897);  Nature, 
Vol.  56,  p.  378. 

1755.  Let  me  mention  one  other  contribution  which  this 
theory  [Theory  of  functions  of  a  complex  variable]  has  made  to 
knowledge  lying  somewhat  outside  our  track.  During  the 
rigorous  revision  to  which  the  foundations  of  the  theory  have 
been  subjected  in  its  re-establishment  by  Weierstrass,  new 
ideas  as  regards  number  and  continuity  have  been  introduced. 
With  him  and  with  others  influenced  by  him,  there  has  thence 
sprung  a  new  theory  of  higher  arithmetic;  and  with  its  growth, 
much  has  concurrently  been  effected  in  the  elucidation  of  the 
general  notions  of  number  and  quantity.  ...  It  thus  appears 
to  be  the  fact  that,  as  with  Plato,  or  Descartes,  or  Leibnitz,  or 
Kant,  the  activity  of  pure  mathematics  is  again  lending  some 
assistance  to  the  better  comprehension  of  those  notions  of 
time,  space,  number,  quantity,  which  underlie  a  philosophical 
conception  of  the  universe. — FORSYTH,  A.  R. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science  (1897);  Nature, 
Vol.  56,  p.  878. 


CHAPTER  XVIII 

GEOMETRY 

1801.  The  science  of  figures  is  most  glorious  and  beautiful. 
But  how  inaptly  it  has  received  the  name  geometry! 

FRISCHLINUS,  N. 
Dialog  1. 

1802.  Plato  said  that  God  geometrizes  continually. 

PLUTARCH. 

Convivialium  disputationum,  liber  8,  2. 


1803.  /ATjSew  a7€&)/ier/37/T05  «Wra>  /AOV  rrjv 

[Let  no  one  ignorant  of  geometry  enter  my  door.] 

PLATO. 

Tzetzes,  Chiliad,  8,  972. 

1804.  All  the  authorities  agree  that  he  [Plato]  made  a  study  of 
geometry  or  some  exact  science  an  indispensable  preliminary  to 
that  of  philosophy.    The  inscription  over  the  entrance  to  his 
school  ran  "Let  none  ignorant  of  geometry  enter  my  door," 
and  on  one  occasion  an  applicant  who  knew  no  geometry  is  said 

to  have  been  refused  admission  as  a  student.  —  BALL,  W.  W.  R. 

History  of  Mathematics  (London,  1901),  p.  45. 

1805.  Form  and  size  constitute  the  foundation  of  all  search 
for  truth.  —  PARKER,  F.  W. 

Talks  on  Pedagogics  (New  York,  1894),  P-  72. 

1806.  At  present  the  science  [of  geometry]  is  hi  flat  contradic- 
tion to  the  language  which  geometricians  use,  as  will  hardly  be 
denied  by  those  who  have  any  acquaintance  with  the  study: 
for  they  speak  of  finding  the  side  of  a  square,  and  applying  and 
adding,  and  so  on,  as  if  they  were  engaged  in  some  business,  and 
as  if  all  their  propositions  had  a  practical  end  in  view:  whereas  in 
reality  the  science  is  pursued  wholly  for  the  sake  of  knowledge. 

292 


GEOMETRY  293 

Certainly,  he  said. 

Then  must  not  a  further  admission  be  made? 

What  admission? 

The  admission  that  this  knowledge  at  which  geometry  aims  is 
of  the  eternal,  and  not  of  the  perishing  and  transient. 

That  may  be  easily  allowed.  Geometry,  no  doubt,  is  the 
knowledge  of  what  eternally  exists. 

Then,  my  noble  friend,  geometry  will  draw  the  soul  towards 
truth,  and  create  the  mind  of  philosophy,  and  raise  up  that 
which  is  now  unhappily  allowed  to  fall  down. — PLATO. 

Republic  [Jowett-Davies],  Bk.  7,  p.  527. 

1807.  Among  them  [the  Greeks]  geometry  was  held  in  highest 
honor:  nothing  was  more  glorious  than  mathematics.    But  we 
have  limited  the  usefulness  of  this  art  to  measuring  and  calcu- 
lating.— CICERO  . 

Tusculanae  Disputationes,  1,  2,5. 

1808.  Geometria, 
Through  which  a  man  hath  the  sleight 
Of  length,  and  brede,  of  depth,  of  height. 

GOWER,  JOHN. 

Confessio  Amantis,  Bk.  7. 

1809.  Geometrical  truths  are  in  a  way  asymptotes  to  physical 
truths,  that  is  to  say,  the  latter  approach  the  former  indefinitely 
near  without  ever  reaching  them  exactly. — D'ALEMBERT. 

Quoted  in  Rebiere:  Mathematiques  et  Mathe- 
maticiens  (Paris,  1898),  p.  10. 

1810.  Geometry  exhibits  the  most  perfect  example  of  logical 
stratagem. — BUCKLE,  H.  T. 

History  of  Civilization  in  England  (New  York, 
1891),  Vol.  2,  p.  342. 

1811.  It  is  the  glory  of  geometry  that  from  so  few  principles, 
fetched  from  without,  it  is  able  to  accomplish  so  much. 

NEWTON. 

Philosophiae  Naturalis  Principia  Mathemat- 
ica,  Praefat. 


294  MEMORABILIA    MATHEMATICA 

1812.  Geometry  is  the  application  of  strict  logic  to  those 
properties  of  space  and  figure  which  are  self-evident,  and  which 
therefore  cannot  be  disputed.    But  the  rigor  of  this  science  is 
carried  one  step  further;  for  no  property,  however  evident  it 
may  be,  is  allowed  to  pass  without  demonstration,  if  that  can  be 
given.    The  question  is  therefore  to  demonstrate  all  geometrical 
truths  with  the  smallest  possible  number  of  assumptions. 

DE  MORGAN,  A. 

On  the  Study  and  Difficulties  of  Mathematics 
(Chicago ,1902],  p.  231. 

1813.  Geometry  is  a  true  natural  science : — only  more  simple, 
and  therefore  more  perfect  than  any  other.    We  must  not  sup- 
pose that,  because  it  admits  the  application  of  mathematical 
analysis,  it  is  therefore  a  purely  logical  science,  independent  of 
observation.    Every  body  studied  by  geometers  presents  some 
primitive  phenomena  which,  not  being  discoverable  by  reason- 
ing, must  be  due  to  observation  alone. — COMTE,  A. 

Positive  Philosophy  [Martineau],  Bk.  1,  chap.  3. 

1814.  Geometry  in   every  proposition   speaks   a   language 
which  experience  never  dares  to  utter;  and  indeed  of  which  she 
but  half  comprehends  the  meaning.    Experience  sees  that  the 
assertions  are  true,  but  she  sees  not  how  profound  and  absolute 
is  their  truth.     She  unhesitatingly  assents  to  the  laws  which 
geometry  delivers,  but  she  does  not  pretend  to  see  the  origin  of 
their  obligation.    She  is  always  ready  to  acknowledge  the  sway 
of  pure  scientific  principles  as  a  matter  of  fact,  but  she  does  not 
dream  of  offering  her  opinion  on  their  authority  as  a  matter  of 
right;  still  less  can  she  justly  claim  to  herself  the  source  of  that 
authority. — WHEWELL,  WILLIAM. 

The   Philosophy   of  the    Inductive   Sciences, 
Part  1,  Bk.  1,  chap.  6,  sect.  1  (London,  1858). 

1815.  Geometry  is  the  science  created  to  give  understanding 
and  mastery  of  the  external  relations  of  things;  to  make  easy 
the  explanation  and  description  of  such  relations  and  the 
transmission  of  this  mastery. — HALSTED,  G.  B. 

Proceedings  of  the  American  Association  for 
the  Advancement  of  Science  (1904),  P-  359. 


GEOMETRY  295 

1816.  A  mathematical  point  is  the  most  indivisible  and 
unique  thing  which  art  can  present. — DONNE,  JOHN. 

Letters,  21. 

1817.  It  is  certain  that  from  its  completeness,  uniformity  and 
faultlessness,  from  its  arrangement  and  progressive  character, 
and  from  the  universal  adoption  of  the  completest  and  best  line 
of  argument,  Euclid's  "  Elements  "  stand  preeminently  at  the 
head  of  all  human  productions.    In  no  science,  in  no  department 
of  knowledge,  has  anything  appeared  like  this  work:  for  upward 
of  2000  years  it  has  commanded  the  admiration  of  mankind,  and 
that  period  has  suggested  little  toward  its  improvement. 

KELLAND,  P. 

Lectures  on  the  Principles  of  Demonstrative 
Mathematics  (London,  1848),  p.  17. 

1818.  In  comparing  the  performance  in  Euclid  with  that 
in   Arithmetic   and   Algebra   there   could  be  no  doubt  that 
Euclid  had  made  the  deepest  and  most  beneficial  impression: 
in  fact  it  might  be  asserted  that  this  constituted  by  far  the  most 
valuable  part  of  the  whole  training  to  which  such  persons 
[students,  the  majority  of  which  were  not  distinguished  for 
mathematical  taste  and  power]  were  subjected. 

TODHUNTER,    I. 

Essay  on  Elementary  Geometry;  Conflict  of 
Studies  and  other  Essays  (London,  1878), 
p.  167. 

1819.  In  England  the  geometry  studied  is  that  of  Euclid,  and 
I  hope  it  never  will  be  any  other;  for  this  reason,  that  so  much 
has  been  written  on  Euclid,  and  all  the  difficulties  of  geometry 
have  so  uniformly  been  considered  with  reference  to  the  form 
in  which  they  appear  in  Euclid,  that  the  study  of  that  author  is  a 
better  key  to  a  great  quantity  of  useful  reading  than  any  other. 

DE  MORGAN,  A. 

Elements  of  Algebra  (London,  1887),  Introduc- 
tion. 

1820.  This  book  [Euclid]  has  been  for  nearly  twenty-two 
centuries  the  encouragement  and  guide  of  that  scientific  thought 


296  MEMORABILIA   MATHEMATICA 

which  is  one  thing  with  the  progress  of  man  from  a  worse  to  a 
better  state.  The  encouragement;  for  it  contained  a  body  of 
knowledge  that  was  really  known  and  could  be  relied  on,  and 
that  moreover  was  growing  hi  extent  and  application.  For 
even  at  the  tune  this  book  was  written — shortly  after  the 
foundation  of  the  Alexandrian  Museum — Mathematics  was  no 
longer  the  merely  ideal  science  of  the  Platonic  school,  but  had 
started  on  her  career  of  conquest  over  the  whole  world  of 
Phenomena.  The  guide;  for  the  aim  of  every  scientific  student 
of  every  subject  was  to  bring  his  knowledge  of  that  subject  into 
a  form  as  perfect  as  that  which  geometry  had  attained.  Far 
up  on  the  great  mountain  of  Truth,  which  all  the  sciences  hope 
to  scale,  the  foremost  of  that  sacred  sisterhood  was  seen,  beckon- 
ing for  the  rest  to  follow  her.  And  hence  she  was  called,  in  the 
dialect  of  the  Phythagoreans,  "  the  purifier  of  the  reasonable 
soul." — CLIFFORD,  W.  K. 

Lectures  and  Essays  (London,  1901),  Vol.  1, 

p.  854^ 

1821.  [Euclid]   at   once  the  inspiration  and  aspiration  of 
scientific  thought. — CLIFFORD,  W.  K. 

Lectures  and  Essays  (London,  1901),  Vol  1, 
p.  855. 

1822.  The  "elements"  of  the  Great  Alexandrian  remain  for 
all  time  the  first,  and  one  may  venture  to  assert,  the  only  perfect 
model  of  logical  exactness  of  principles,  and  of  rigorous  develop- 
ment of  theorems.    If  one  would  see  how  a  science  can  be  con- 
structed and  developed  to  its  minutest  details  from  a  very  small 
number  of  intuitively  perceived  axioms,  postulates,  and  plain 
definitions,  by  means  of  rigorous,  one  would  almost  say  chaste, 
syllogism,  which  nowhere  makes  use  of  surreptitious  or  foreign 
aids,  if  one  would  see  how  a  science  may  thus  be  constructed 
one  must  turn  to  the  elements  of  Euclid. — HANKEL,  H. 

Die  Entwickelung  der  Mathematik  in  den  letzten 
Jahrhunderten  (Tubingen,  1884),  P-  7. 

1823.  If  we  consider  him  [Euclid]  as  meaning  to  be  what  his 
commentators  have  taken  him  to  be,  a  model  of  the  most  un- 


GEOMETRY  297 

scrupulous  formal  rigour,  we  can  deny  that  he  has  altogether 
succeeded,  though  we  admit  that  he  made  the  nearest  approach. 

DE  MORGAN,  A. 

Smith's  Dictionary  of  Greek  and  Roman  Biogra- 
phy and  Mythology  (London,  1902);  Article 
"Eucleides." 

1824.  The  Elements  of  Euclid  is  as  small  a  part  of  mathe- 
matics as  the  Iliad  is  of  literature;  or  as  the  sculpture  of  Phidias 

is  of  the  world's  total  art. — KEYSER,  C.  J. 

Lectures  on  Science,  Philosophy  and  Art  (New 
York,  1908),  p.  8. 

1825.  I  should  rejoice  to  see  .  .  .  Euclid  honourably  shelved 
or  buried  "  deeper  than  did  ever  plummet  sound  "  out  of  the 
schoolboys'  reach;  morphology  introduced  into  the  elements  of 
algebra;  projection,  correlation,  and  motion  accepted  as  aids  to 
geometry;  the  mind  of  the  student  quickened  and  elevated  and 
his  faith  awakened  by  early  initiation  into  the  ruling  ideas  of 
polarity,    continuity,    infinity,    and   familiarization   with   the 
doctrines  of  the  imaginary  and  inconceivable. 

SYLVESTER,  J.  J. 

A  Plea  for  the  Mathematician;  Nature,  Vol.  1 , 
p.  261. 

1826.  The  early  study  of  Euclid  made  me  a  hater  of  geom- 
etry, .  .  .  and  yet,  in  spite  of  this  repugnance,  which  had 
become  a  second  nature  hi  me,  whenever  I  went  far  enough  into 
any  mathematical  question,  I  found  I  touched,  at  last,  a  geo- 
metrical bottom. — SYLVESTER,  J.  J. 

A  Plea  for  the  Mathematician;  Nature,  Vol.  1, 
p.  262. 

1827.  Newton  had  so  remarkable  a  talent  for  mathematics 
that  Euclid's  Geometry  seemed  to  him  "a  trifling  book,"  and  he 
wondered  that  any  man  should  have  taken  the  trouble  to 
demonstrate  propositions,  the  truth  of  which  was  so  obvious  to 
him  at  the  first  glance.    But,  on  attempting  to  read  the  more 
abstruse  geometry  of  Descartes,  without  having  mastered  the 
elements  of  the  science,  he  was  baffled,  and  was  glad  to  come 
back  again  to  his  Euclid. — PARTON,  JAMES. 

Sir  Isaac  Newton. 


298  MEMORABILIA   MATHEMATICA 

1828.  As  to  the  need  of  improvement  there  can  be  no  ques- 
tion whilst  the  reign  of  Euclid  continues.    My  own  idea  of  a 
useful  course  is  to  begin  with  arithmetic,  and  then  not  Euclid 
but  algebra.    Next,  not  Euclid,  but  practical  geometry,  solid  as 
well  as  plane;  not  demonstration,  but  to  make  acquaintance. 
Then  not  Euclid,  but  elementary  vectors,  conjoined  with  alge- 
bra, and  applied  to  geometry.    Addition  first;  then  the  scalar 
product.    Elementary  calculus  should  go  on  simultaneously,  and 
come  into  the  vector  algebraic  geometry  after  a  bit.     Euclid 
might  be  an  extra  course  for  learned  men,  like  Homer.    But  Eu- 
clid for  children  is  barbarous. — HEAVISIDE,  OLIVER. 

Electro-Magnetic  Theory  (London,  1893), 
Vol.  1,  p.  148. 

1829.  Geometry  is  nothing  if  it  be  not  rigorous,  and  the 
whole  educational  value  of  the  study  is  lost,  if  strictness  of 
demonstration  be  trifled  with.    The  methods  of  Euclid  are,  by 
almost  universal  consent,  unexceptionable  in  point  of  rigour. 

SMITH,  H.  J.  S. 
Nature,  Vol.  8,  p.  450. 

1830.  To  seek  for  proof  of  geometrical  propositions  by  an 
appeal  to  observation  proves  nothing  in  reality,  except  that  the 
person  who  has  recourse  to  such  grounds  has  no  due  apprehen- 
sion of  the  nature  of  geometrical  demonstration.     We  have 
heard  of  persons  who  convince  themselves  by  measurement  that 
the  geometrical  rule  respecting  the  squares  on  the  sides  of  a 
right-angles  triangle  was  true:  but  these  were  persons  whose 
minds  had  been  engrossed  by  practical  habits,  and  in  whom 
speculative  development  of  the  idea  of  space  had  been  stifled 
by  other  employments. — WHEWELL,  WILLIAM. 

The  Philosophy  of  the  Inductive  Sciences, 
(London,  1858),  Part  1,  Bk.  2,  chap.  1,  sect.  4- 

1831.  No  one  has  ever  given  so  easy  and  natural  a  chain  of 
geometrical  consequences  [as  Euclid].    There  is  a  never-erring 
truth  in  the  results. — DE  MORGAN,  A. 

Smith's  Dictionary  of  Greek  and  Roman  Biog- 
raphy and  Mythology  (London,  1902};  Article 
"  Eucleides." 


GEOMETRY  299 

1832.  Beyond  question,  Egyptian  geometry,  such  as  it  was, 
was  eagerly  studied  by  the  early  Greek  philosophers,  and  was 
the  germ  from  which  in  their  hands  grew  that  magnificent  science 
to  which  every  Englishman  is  indebted  for  his  first  lessons  in 
right  seeing  and  thinking. — Gow,  JAMES. 

A  Short  History  of  Greek  Mathematics  (Cam- 
bridge, 1884),  p.  181. 

1833.  A  figure  and  a  step  onward : 
Not  a  figure  and  a  florin. 

MOTTO  OF  THE  PYTHAGOREAN  BROTHERHOOD. 

W.  B.  Frankland:  Story  of  Euclid  (London, 
1902},  p.  38. 

1834.  The  doctrine  of  proportion,  as  laid  down  in  the  fifth 
book  of  Euclid,  is,  probably,  still  unsurpassed  as  a  masterpiece 
of  exact  reasoning;  although  the  cumbrousness  of  the  forms  of 
expression  which  were  adopted  in  the  old  geometry  has  led  to  the 
total  exclusion  of  this  part  of  the  elements  from  the  ordinary 
course  of  geometrical  education.    A  zealous  defender  of  Euclid 
might  add  with  truth  that  the  gap  thus  created  hi  the  elemen- 
tary teaching  of  mathematics  has  never  been  adequately  sup- 
plied.— SMITH,  H.  J.  S. 

Presidential  Address  British  Association  for 
the  Advancement  of  Science  (1873);  Nature, 
Vol.  8,  p.  451. 

1835.  The  Definition  in  the  Elements,  according  to  Clavius, 
is  this:  Magnitudes  are  said  to  be  in  the  same  Reason  [ratio],  a 
first  to  a  second,  and  a  third  to  a  fourth,  when  the  Equimulti- 
ples of  the  first  and  third  according  to  any  Multiplication 
whatsoever  are  both  together  either  short  of,  equal  to,  or  exceed 
the  Equimultiples  of  the  second  and  fourth,  if  those  be  taken, 
which  answer  one  another.  .  .  .  Such  is  Euclid's  Definition  of 
Proportions;  that  scare-Crow  at  which  the  over  modest  or 
slothful  Dispositions  of  Men  are  generally  affrighted:  they  are 
modest,  who  distrust  their  own  Ability,  as  soon  as  a  Difficulty 
appears,  but  they  are  slothful  that  will  not  give  some  Attention 
for  the  learning  of  Sciences;  as  if  while  we  are  involved  in 
Obscurity  we  could  clear  ourselves  without  Labour.    Both  of 


300  MEMORABILIA   MATHEMATICA 

which  Sorts  of  Persons  are  to  be  admonished,  that  the  former  be 
not  discouraged,  nor  the  latter  refuse  a  little  Care  and  Diligence 
when  a  Thing  requires  some  Study. — BARROW,  ISAAC. 

Mathematical  Lectures  (London,  1734),  P-  388. 

1836.  Of  all  branches  of  human  knowledge,  there  is  none 
which,  like  it  [geometry]  has  sprung  a  completely  armed  Minerva 
from  the  head  of  Jupiter;  none  before  whose  death-dealing 
Aegis  doubt  and  inconsistency  have  so  little  dared  to  raise  their 
eyes.    It  escapes  the  tedious  and  troublesome  task  of  collecting 
experimental  facts,  which  is  the  province  of  the  natural  sciences 
in  the  strict  sense  of  the  word:  the  sole  form  of  its  scientific 
method  is  deduction.    Conclusion  is  deduced  from  conclusion, 
and  yet  no  one  of  common  sense  doubts  but  that  these  geo- 
metrical principles  must  find  their  practical  application  in  the 
real  world  about  us.    Land  surveying,  as  well  as  architecture, 
the   construction   of   machinery   no   less   than   mathematical 
physics,  are  continually  calculating  relations  of  space  of  the 
most  varied  kinds  by  geometrical  principles;  they  expect  that 
the  success  of  their  constructions  and  experiments  shall  agree 
with  their  calculations;  and  no  case  is  known  in  which  this 
expectation  has  been  falsified,  provided  the  calculations  were 
made  correctly  and  with  sufficient  data. — HELMHOLTZ,  H. 

The  Origin  and  Significance  of  Geometrical 
Axioms; Popular  Scientific  Lectures  [Atkinson], 
Second  Series  (New  York,  1881),  p.  27. 

1837.  The  amazing  triumphs  of  this  branch  of  mathematics 
[geometry]  show  how  powerful  a  weapon  that  form  of  deduction 
is  which  proceeds  by  an  artificial  separation  of  facts,  in  them- 
selves inseparable. — BUCKLE,  H.  T. 

History  of  Civilization  in  England  (New  York, 
1891),  Vol.  2,  p.  343. 

1838.  Every  theorem  in  geometry  is  a  law  of  external  nature, 
and  might  have  been  ascertained  by  generalizing  from  observa- 
tion and  experiment,  which  hi  this  case  resolve  themselves  into 
comparisons  and  measurements.    But  it  was  found  practicable, 
and  being  practicable  was  desirable,  to  deduce  these  truths  by 
ratiocination  from  a  small  number  of  general  laws  of  nature,  the 
certainty  and  universality  of  which  was  obvious  to  the  most 


GEOMETRY  301 

careless  observer,  and  which  compose  the  first  principles  and 
ultimate  premises  of  the  science. — MILL,  J.  S. 

System  of  Logic,  Bk.  3,  chap.  24,  sect.  7. 

1839.  All  such  reasonings   [natural  philosophy,   chemistry, 
agriculture,  political  economy,  etc.]  are,  in  comparison  with 
mathematics,  very  complex;  requiring  so  much  more  than  that 
does,  beyond  the  process  of  merely  deducing  the  conclusion 
logically  from  the  premises:  so  that  it  is  no  wonder  that  the 
longest  mathematical  demonstration   should  be  much  more 
easily  constructed  and  understood,  than  a  much  shorter  train  of 
just  reasoning  concerning  real  facts.     The  former  has  been 
aptly  compared  to  a  long  and  steep,  but  even  and  regular, 
flight  of  steps,  which  tries  the  breath,  and  the  strength,  and  the 
perseverance  only;  while  the  latter  resembles  a  short,  but  rug- 
ged and  uneven,  ascent  up  a  precipice,  which  requires  a  quick 
eye,  agile  limbs,  and  a  firm  step;  and  in  which  we  have  to  tread 
now  on  this  side,  now  on  that — ever  considering  as  we  proceed, 
whether  this  or  that  projection  will  afford  room  for  our  foot, 
or  whether  some  loose  stone  may  not  slide  from  under  us.    There 
are  probably  as  many  steps  of  pure  reasoning  in  one  of  the  longer 
of  Euclid's  demonstrations,  as  in  the  whole  of  an  argumentative 
treatise  on  some  other  subject,  occupying  perhaps  a  considerable 
volume. — WHATELY,  R. 

Elements  of  Logic,  Bk.  4>  chap.  2,  sect.  5. 

1840.  [Geometry]  that  held  acquaintance  with  the  stars, 
And  wedded  soul  to  soul  in  purest  bond 

Of  reason,  undisturbed  by  space  or  time. 

WORDSWORTH. 
The  Prelude,  Bk.  5. 

1841.  The  statement  that  a  given  individual  has  received  a 
sound  geometrical  training  implies  that  he  has  segregated  from 
the  whole  of  his  sense  impressions  a  certain  set  of  these  impres- 
sions, that  he  has  eliminated  from  their  consideration  all  ir- 
relevant impressions  (in  other  words,  acquired  a  subjective 
command  of  these  impressions),  that  he  has  developed  on  the 
basis  of  these  impressions  an  ordered  and  continuous  system  of 
logical  deduction,  and  finally  that  he  is  capable  of  expressing  the 


302  MEMORABILIA   MATHEMATICA 

nature  of  these  impressions  and  his  deductions  therefrom  in 
terms  simple  and  free  from  ambiguity.  Now  the  slightest 
consideration  will  convince  any  one  not  already  conversant  with 
the  idea,  that  the  same  sequence  of  mental  processes  underlies 
the  whole  career  of  any  individual  hi  any  walk  of  life  if  only  he  is 
not  concerned  entirely  with  manual  labor;  consequently  a  full 
training  in  the  performance  of  such  sequences  must  be  regarded 
as  forming  an  essential  part  of  any  education  worthy  of  the 
name.  Moreover  the  full  appreciation  of  such  processes  has  a 
higher  value  than  is  contained  in  the  mental  training  involved, 
great  though  this  be,  for  it  induces  an  appreciation  of  intellectual 
unity  and  beauty  which  plays  for  the  mind  that  part  which  the 
appreciation  of  schemes  of  shape  and  color  plays  for  the  artistic 
faculties;  or,  again,  that  part  which  the  appreciation  of  a  body  of 
religious  doctrine  plays  for  the  ethical  aspirations.  Now  geom- 
etry is  not  the  sole  possible  basis  for  inculcating  this  apprecia- 
tion. Logic  is  an  alternative  for  adults,  provided  that  the 
individual  is  possessed  of  sufficient  wide,  though  rough,  expe- 
rience on  which  to  base  his  reasoning.  Geometry  is,  however, 
highly  desirable  in  that  the  objective  bases  are  so  simple  and 
precise  that  they  can  be  grasped  at  an  early  age,  that  the 
amount  of  training  for  the  imagination  is  very  large,  that  the 
deductive  processes  are  not  beyond  the  scope  of  ordinary  boys, 
and  finally  that  it  affords  a  better  basis  for  exercise  in  the  art  of 
simple  and  exact  expression  than  any  other  possible  subject  of  a 

school  course. — CARSON,  G.  W.  L. 

The  Functions  of  Geometry  as  a  Subject  of  Edu- 
cation (Tonbridge,  1910),  p.  8. 

1842.  It  seems  to  me  that  the  thing  that  is  wanting  in  the 
education  of  women  is  not  the  acquaintance  with  any  facts,  but 
accurate  and  scientific  habits  of  thought,  and  the  courage  to 
think  that  true  which  appears  unlikely.  And  for  supplying  this 
want  there  is  a  special  advantage  in  geometry,  namely  that  it 
does  not  require  study  of  a  physically  laborious  kind,  but  rather 
that  rapid  intuition  which  women  certainly  possess;  so  that  it  is 
fit  to  become  a  scientific  pursuit  for  them. — CLIFFORD,  W.  K. 

Quoted  by  Pollock  in  Clifford's  Lectures  and 
Essays  (London,  1901),  Vol.  1,  Introduction, 
p.  43. 


GEOMETRY  303 

1843.  On  the  lecture  slate 

The  circle  rounded  under  female  hands 
With  flawless  demonstration. — TENNYSON. 

The  Princess,  II,  L  493. 

1844.  It  is  plain  that  that  part  of  geometry  which  bears  upon 
strategy  does  concern  us.    For  in  pitching  camps,  or  in  occupy- 
ing positions,  or  in  closing  or  extending  the  lines  of  an  army,  and 
in  all  the  other  manoeuvres  of  an  army  whether  in  battle  or  on 
the  march,  it  will  make  a  great  difference  to  a  general,  whether 
he  is  a  geometrician  or  not. — PLATO. 

Republic,  Bk.  7,  p.  526. 

1845.  Then  nothing  should  be  more  effectually  enacted,  than 
that  the  inhabitants  of  your  fair  city  should  learn  geometry. 
Moreover  the  science  has  indirect  effects,  which  are  not  small. 

Of  what  kind  are  they?  he  said. 

There  are  the  military  advantages  of  which  you  spoke,  I 
said;  and  in  all  departments  of  study,  as  experience  proves,  any 
one  who  has  studied  geometry  is  infinitely  quicker  of  apprehen- 
sion.— PLATO. 

Republic  [Jowett],  Bk.  7,  p.  527. 

1846.  It  is  doubtful  if  we  have  any  other  subject  that  does 
so  much  to  bring  to  the  front  the  danger  of  carelessness,  of 
slovenly  reasoning,  of  inaccuracy,  and  of  forgetfulness  as  this 
science  of  geometry,  which  has  been  so  polished  and  perfected  as 

the  centuries  have  gone  on. — SMITH,  D.  E. 

The  Teaching  of  Geometry  (Boston,  1911},  p.  12. 

1847.  The  culture  of  the  geometric  imagination,  tending  to 
produce  precision  in  remembrance  and  invention  of  visible  forms 
will,  therefore,  tend  directly  to  increase  the  appreciation  of 
works  of  belles-letters. — HILL,  THOMAS. 

The    Uses   of   Mathesis;   Bibliotheca   Sacra, 
Vol.  32,  p.  504. 

1848.  Yet  may  we  not  entirely  overlook 

The  pleasures  gathered  from  the  rudiments 
Of  geometric  science.    Though  advanced 
In  these  inquiries,  with  regret  I  speak, 


304  MEMORABILIA   MATHEMATICA 

No  farther  than  the  threshold,  there  I  found 

Both  elevation  and  composed  delight: 

With  Indian  awe  and  wonder,  ignorance  pleased 

With  its  own  struggles,  did  I  meditate 

On  the  relations  those  abstractions  bear 

To  Nature's  laws. 

More  frequently  from  the  same  source  I  drew 

A  pleasure  quiet  and  profound,  a  sense 

Of  permanent  and  universal  sway, 

And  paramount  belief;  there,  recognized 

A  type,  for  finite  natures,  of  the  one 

Supreme  Existence,  the  surpassing  life 

Which  to  the  boundaries  of  space  and  time, 

Of  melancholy  space  and  doleful  time, 

Superior  and  incapable  of  change, 

Nor  touched  by  welterings  of  passion — is, 

And  hath  the  name  of  God.    Transcendent  peace 

And  silence  did  wait  upon  these  thoughts 

That  were  a  frequent  comfort  to  my  youth. 

Mighty  is  the  charm 
Of  those  abstractions  to  a  mind  beset 
With  images  and  haunted  by  himself, 
And  specially  delightful  unto  me 
Was  that  clear  synthesis  built  up  aloft 
So  gracefully;  even  then  when  it  appeared 
Not  more  than  a  mere  plaything,  or  a  toy 
To  sense  embodied :  not  the  thing  it  is 
In  verity,  an  independent  world, 
Created    out    of    pure    intelligence. — WORDSWORTH. 

The  Prelude,  Bk.  6. 

1849.  'Tis  told  by  one  whom  stormy  waters  threw, 
With  fellow-sufferers  by  the  shipwreck  spared, 
Upon  a  desert  coast,  that  having  brought 
To  land  a  single  volume,  saved  by  chance, 
A  treatise  of  Geometry,  he  wont, 


GEOMETRY  305 

Although  of  food  and  clothing  destitute, 
And  beyond  common  wretchedness  depressed, 
To  part  from  company,  and  take  this  book 
(Then  first  a  self  taught  pupil  in  its  truths) 
To  spots  remote,  and  draw  his  diagrams 
With  a  long  staff  upon  the  sand,  and  thus 
Did  oft  beguile  his  sorrow,  and  almost 
Forget  his  feeling: — WORDSWORTH. 

The  Prelude,  Bk.  6. 

1850.  We  study  art  because  we  receive  pleasure  from  the 
great  works  of  the  masters,  and  probably  we  appreciate  them 
the  more  because  we  have  dabbled  a  little  in  pigments  or  in 
clay.  We  do  not  expect  to  be  composers,  or  poets,  or  sculptors, 
but  we  wish  to  appreciate  music  and  letters  and  the  fine  arts,  and 
to  derive  pleasure  from  them  and  be  uplifted  by  them.  .  .  . 

So  it  is  with  geometry.  We  study  it  because  we  derive  pleas- 
ure from  contact  with  a  great  and  ancient  body  of  learning  that 
has  occupied  the  attention  of  master  minds  during  the  thousands 
of  years  in  which  it  has  been  perfected,  and  we  are  uplifted  by 
it.  To  deny  that  our  pupils  derive  this  pleasure  from  the  study 
is  to  confess  ourselves  poor  teachers,  for  most  pupils  do  have 
positive  enjoyment  in  the  pursuit  of  geometry,  in  spite  of  the 
tradition  that  leads  them  to  proclaim  a  general  dislike  for  all 
study.  This  enjoyment  is  partly  that  of  the  game, — the  playing 
of  a  game  that  can  always  be  won,  but  that  cannot  be  won  too 
easily.  It  is  partly  that  of  the  aesthetic,  the  pleasure  of  sym- 
metry of  form,  the  delight  of  fitting  things  together.  But 
probably  it  lies  chiefly  in  the  mental  uplift  that  geometry  brings, 
the  contact  with  absolute  truth,  and  the  approach  that  one 
makes  to  the  Infinite.  We  are  not  quite  sure  of  any  one  thing  in 
biology;  our  knowledge  of  geology  is  relatively  very  slight,  and 
the  economic  laws  of  society  are  uncertain  to  every  one  except 
some  individual  who  attempts  to  set  them  forth;  but  before  the 
world  was  fashioned  the  square  on  the  hypotenuse  was  equal  to 
the  sum  of  the  squares  on  the  other  two  sides  of  a  right  triangle, 
and  it  will  be  so  after  this  world  is  dead;  and  the  inhabitant  of 
Mars,  if  he  exists,  probably  knows  its  truth  as  we  know  it.  The 
uplift  of  this  contact  with  absolute  truth,  with  truth  eternal, 


306  MEMORABILIA   MATHEMATICA 

gives  pleasure  to  humanity  to  a  greater  or  less  degree,  depending 
upon  the  mental  equipment  of  the  particular  individual;  but  it 
probably  gives  an  appreciable  amount  of  pleasure  to  every 
student  of  geometry  who  has  a  teacher  worthy  of  the  name. 

SMITH,  D.  E. 
The  Teaching  of  Geometry  (Boston,  1911),  p.  16. 

1851.  No  other  person  can  judge  better  of  either  [the  merits  of 
a  writer  and  the  merits  of  his  works]  than  himself;  for  none  have 
had  access  to  a  closer  or  more  deliberate  examination  of  them. 
It  is  for  this  reason,  that  in  proportion  that  the  value  of  a  work 
is  intrinsic,  and  independent  of  opinion,  the  less  eagerness  will 
the  author  feel  to  conciliate  the  suffrages  of  the  public.    Hence 
that  inward  satisfaction,  so  pure  and  so  complete,  which  the 
study  of  geometry  yields.     The  progress  which  an  individual 
makes  in  this  science,  the  degree  of  eminence  which  he  attains 
in  it,  all  this  may  be  measured  with  the  same  rigorous  accuracy 
as  the  methods  about  which  his  thoughts  are  employed.    It  is 
only  when  we  entertain  some  doubts  about  the  justness  of  our 
own  standard,  that  we  become  anxious  to  relieve  ourselves  from 
our  uncertainty,  by  comparing  it  with  the  standard  of  another. 
Now,  in  all  matters  which  fall  under  the  cognizance  of  taste,  this 
standard  is  necessarily  somewhat  variable;  depending  on  a 
sort  of  gross  estimate,  always  a  little  arbitrary,  either  in  whole 
or  hi  part;  and  liable  to  continual  alteration  in  its  dimensions, 
from  negligence,  temper,  or  caprice.    In  consequence  of  these 
circumstances  I  have  no  doubt,  that  if  men  lived  separate  from 
each  other,  and  could  in  such  a  situation  occupy  themselves 
about  anything  but  self-preservation,  they  would  prefer  the 
study  of  the  exact  sciences  to  the  cultivation  of  the  agreeable 
arts.    It  is  chiefly  on  account  of  others,  that  a  man  aims  at 
excellence  hi  the  latter,  it  is  on  his  own  account  that  he  devotes 
himself  to  the  former.    In  a  desert  island,  accordingly,  I  should 
think  that  a  poet  could  scarcely  be  vain;  whereas  a  geometrician 
might  still  enjoy  the  pride  of  discovery. — D  'ALEMBERT. 

Essai  sur  les  Gens  Lettres;  Melages  (Amsterdam 
1764),  LI,  p.  384. 

1852.  If  it  were  required  to  determine  inclined  planes  of 
varying  inclinations  of  such  lengths  that  a  free  rolling  body 


GEOMETRY  307 

would  descend  on  them  in  equal  times,  any  one  who  under- 
stands the  mechanical  laws  involved  would  admit  that  this 
would  necessitate  sundry  preparations.  But  in  the  circle  the 
proper  arrangement  takes  place  of  its  own  accord  for  an  infinite 
variety  of  positions  yet  with  the  greatest  accuracy  in  each 
individual  case.  For  all  chords  which  meet  the  vertical  diameter 
whether  at  its  highest  or  lowest  point,  and  whatever  their 
inclinations,  have  this  in  common:  that  the  free  descent  along 
them  takes  place  in  equal  times.  I  remember,  one  bright  pupil, 
who,  after  I  had  stated  and  demonstrated  this  theorem  to  him, 
and  he  had  caught  the  full  import  of  it,  was  moved  as  by  a 
miracle.  And,  indeed,  there  is  just  cause  for  astonishment  and 
wonder  when  one  beholds  such  a  strange  union  of  manifold 
things  in  accordance  with  such  fruitful  rules  in  so  plain  and 
simple  an  object  as  the  circle.  Moreover,  there  is  no  miracle  in 
nature,  which  because  of  its  pervading  beauty  or  order,  gives 
greater  cause  for  astonishment,  unless  it  be,  for  the  reason  that 
its  causes  are  not  so  clearly  comprehended,  marvel  being  a 
daughter  of  ignorance. — KANT. 

Der  einzig  mogliche  Beweisgrund  zu  einer 
Demonstration  des  Daseins  Gottes;  Werke 
(Hartenstein),  Bd.  2,  p.  137. 

1853.  These  examples  [taken  from  the  geometry  of  the  circle] 
indicate  what  a  countless  number  of  other  such  harmonic  rela- 
tions obtain  in  the  properties  of  space,  many  of  which  are  mani- 
fested in  the  relations  of  the  various  classes  of  curves  in  higher 
geometry,  all  of  which,  besides  exercising  the  understanding 
through  intellectual  insight,  affect  the  emotion  in  a  similar  or 
even  greater  degree  than  the  occasional  beauties  of  nature. 

KANT. 

Der  einzig  mogliche  Beweisgrund  zu  einer 
Demonstration  des  Daseins  Gottes;  Werke 
(Hartenstein),  Bd.  2,  p.  138. 

1854.  But  neither  thirty  years,  nor  thirty  centuries,  affect  the 
clearness,  or  the  charm,  of  Geometrical  truths.    Such  a  theorem 
as  "  the  square  of  the  hypotenuse  of  a  right-angled  triangle  is 
equal  to  the  sum  of  the  squares  of  the  sides  "  is  as  dazzlingly 
beautiful  now  as  it  was  in  the  day  when  Pythagoras  first  dis- 


308  MEMORABILIA   MATHEMATICA 

covered  it,  and  celebrated  its  advent,  it  is  said,  by  sacrificing  a 
hecatomb  of  oxen — a  method  of  doing  honor  to  Science  that  has 
always  seemed  to  me  slightly  exaggerated  and  uncalled-for. 
One  can  imagine  oneself,  even  in  these  degenerate  days,  marking 
the  epoch  of  some  brilliant  scientific  discovery  by  inviting  a 
convivial  friend  or  two,  to  join  one  in  a  beefsteak  and  a  bottle  of 
wine.  But  a  hecatomb  of  oxen!  It  would  produce  a  quite  in- 
convenient supply  of  beef. — DODGSON,  C.  L. 

A  New  Theory  of  Parallels  (London,  1895), 
Introduction,  p.  16. 

1856.  After  Pythagoras  discovered  his  fundamental  theorem 
he  sacrificed  a  hecatomb  of  oxen.  Since  that  time  all  dunces  * 
[Ochsen]  tremble  whenever  a  new  truth  is  discovered. — BOERNE. 

Quoted  in  Moszkowski:  Die  unsterbliche  Kiste 
(Berlin,  1908),  p.  18. 

1856.  Vom  Pythagorieschen  Lehrsatz. 

Die  Wahrheit,  sie  besteht  in  Ewigkeit, 
Wenn  erst  die  blode  Welt  ihr  Licht  erkannt : 
Der  Lehrsatz,  nach  Pythagoras  benannt, 
Gilt  heute,  wie  er  gait  hi  seiner  Zeit. 

Ein  Opfer  hat  Pythagoras  geweiht 
Den  Gottern,  die  den  Lichtstrahl  ihm  gesandt; 
Es  thaten  kund,  geschlachtet  und  verbrannt, 
Eih  hundert  Ochsen  seme  Dankbarkeit. 

Die  Ochsen  seit  den  Tage,  wenn  sie  wittern, 
Dass  erne  neue  Wahrheit  sich  enthiille, 
Erheben  ein  unmenschliches  Gebriille; 

Pythagoras  erfiillt  sie  mit  Entsetzen; 
Und  machtlos,  sich  dem  Licht  zu  wiedersetzen, 
Verschiessen  sie  die  Augen  und  erzittern. 

CHAMISSO,  ADELBERT  VON. 

Gedichte,  1885  (Haushenbusch),  (Berlin,  1889), 
p.  302. 

*  In  the  German  vernacular  a  dunce  or  blockhead  is  called  an  ox. 


GEOMETRY  309 

Truth  lasts  throughout  eternity, 
.  When  once  the  stupid  world  its  light  discerns : 
The  theorem,  coupled  with  Pythagoras'  name, 
Holds  true  today,  as 't  did  in  olden  times. 

A  splendid  sacrifice  Pythagoras  brought 
The  gods,  who  blessed  him  with  this  ray  divine; 
A  great  burnt  offering  of  a  hundred  kine, 
Proclaimed  afar  the  sage's  gratitude. 

Now  since  that  day,  all  cattle  [blockheads]  when  they 

scent 

New  truth  about  to  see  the  light  of  day, 
In  frightful  bellowings  manifest  their  dismay; 

Pythagoras  fills  them  all  with  terror; 
And  powerless  to  shut  out  light  by  error, 
In  sheer  despair  they  shut  their  eyes  and  tremble. 

1857.  To  the  question  "  Which  is  the  signally  most  beautiful 
of  geometrical  truths?"     Frankland  replies:  "One  star  excels 
another  in  brightness,  but  the  very  sun  will  be,  by  common  con- 
sent, a  property  of  the  circle  [Euclid,  Book  3,  Proposition  31] 
selected  for  particular  mention  by  Dante,  that  greatest  of  all 
exponents  of  the  beautiful." — FRANKLAND,  W.  B. 

The  Story  of  Euclid  (London,  1902\  p.  70. 

1858.  As  one 
Who  vers'd  in  geometric  lore,  would  fain 
Measure  the  circle;  and,  though  pondering  long 
And  deeply,  that  beginning,  which  he  needs, 
Finds  not;  e'en  such  was  I,  intent  to  scan 

The  novel  wonder,  and  trace  out  the  form, 

How  to  the  circle  fitted,  and  therein 

How  plac'd:  but  the  flight  was  not  for  my  wing; 

DANTE. 
Paradise  [Carey]  Canto  88,  lines  122-129. 

1859.  If  geometry  were  as  much  opposed  to  our  passions  and 
present  interests  as  is  ethics,  we  should  contest  it  and  violate  it 


310  MEMORABILIA   MATHEMATICA 

but  little  less,  notwithstanding  all  the  demonstrations  of  Euclid 
and  of  Archimedes,  which  you  would  call  dreams  and  believe 
full  of  paralogisms;  and  Joseph  Scaliger,  Hobbes,  and  others, 
who  have  written  against  Euclid  and  Archimedes,  would  not 
find  themselves  in  such  a  small  company  as  at  present. 

LEIBNITZ. 

New  Essays  concerning  Human  Understand- 
ing [Langley],  Bk.  1,  chap.  2,  sect.  12. 

1860.  I  have  no  fault  to  find  with  those  who  teach  geometry. 
That  science  is  the  only  one  which  has  not  produced  sects;  it  is 
founded  on  analysis  and  on  synthesis  and  on  the  calculus;  it 
does  not  occupy  itself  with  probable  truth;  moreover  it  has  the 
same  method  in  every  country. — FREDERICK  THE  GREAT. 

Oeuvres  (Decker),  t.  7,  p.  100. 

1861.  There  are,  undoubtedly,  the  most  ample  reasons  for 
stating  both  the  principles  and  theorems  [of  geometry]  hi  their 
general  form,  .  .  .  But,  that  an  unpractised  learner,  even  in 
making  use  of  one  theorem  to  demonstrate  another,  reasons 
rather  from  particular  to  particular  than  from  the  general 
proposition,  is  manifest  from  the  difficulty  he  finds  in  applying 
a  theorem  to  a  case  in  which  the  configuration  of  the  diagram  is 
extremely  unlike  that  of  the  diagram  by  which  the  original 
theorem  was  demonstrated.    A  difficulty  which,  except  in  cases 
of  unusual  mental  powers,  long  practice  can  alone  remove,  and 
removes  chiefly  by  rendering  us  familiar  with  all  the  configura- 
tions consistent  with  the  general  conditions  of  the  theorem. 

MILL,  J.  S. 
System  of  Logic,  Bk.  2,  chap.  8,  sect.  8. 

1862.  The  reason  why  I  impute  any  defect  to  geometry,  is, 
because  its  original  and  fundamental  principles  are  deriv'd 
merely  from  appearances;  and  it  may  perhaps  be  imagin'd,  that 
this  defect  must  always  attend  it,  and  keep  it  from  ever  reaching 
a  greater  exactness  hi  the  comparison  of  objects  or  ideas,  than 
what  our  eye  or  imagination  alone  is  able  to  attain.    I  own  that 
this  defect  so  far  attends  it,  as  to  keep  it  from  ever  aspiring  to  a 
full  certainty.    But  since  these  fundamental  principles  depend 
on  the  easiest  and  least  deceitful  appearances,  they  bestow  on 


GEOMETRY  311 

their  consequences  a  degree  of  exactness,  of  which  these  conse- 
quences are  singly  incapable. — HUME,  D. 

A  Treatise  of  Human  Nature,  Part  3,  sect.  1 . 

1863.  I  have  already  observed,  that  geometry,  or  the  art,  by 
which  we  fix  the  proportions  of  figures,  tho'  it  much  excels  both 
in  universality  and  exactness,  the  loose  judgments  of  the  senses 
and  imagination;  yet  never  attains  a  perfect  precision  and 
exactness.    Its  first  principles  are  still  drawn  from  the  general 
appearance  of  the  objects;  and  that  appearance  can  never 
afford  us  any  security,  when  we  examine  the  prodigious  minute- 
ness of  which  nature  is  susceptible.  .  .  . 

There  remain,  therefore,  algebra  and  arithmetic  as  the  only 
sciences,  hi  which  we  can  carry  on  a  chain  of  reasoning  to  any 
degree  of  intricacy,  and  yet  preserve  a  perfect  exactness  and 
certainty. — HUME,  D. 

A  Treatise  of  Human  Nature,  Part  S,  sect.  1 . 

1864.  All  geometrical  reasoning  is,  in  the  last  resort,  circular: 
if  we  start  by  assuming  points,  they  can  only  be  defined  by  the 
lines  or  planes  which  relate  them;  and  if  we  start  by  assuming 
lines  or  planes,  they  can  only  be  defined  by  the  points  through 
which  they  pass. — RUSSELL,  BERTRAND. 

Foundations  of  Geometry  (Cambridge,  1897), 
p.  120. 

1865.  The  description  of  right  lines  and  circles,  upon  which 
Geometry  is  founded,  belongs  to  Mechanics.    Geometry  does 
not  teach  us  to  draw  these  lines,  but  requires  them  to  be 
drawn.  ...  it  requires  that  the  learner  should  first  be  taught 
to  describe  these  accurately,  before  he  enters  upon  Geometry; 
then  it  shows  how  by  these  operations  problems  may  be  solved. 
To  describe  right  lines  and  circles  are  problems,  but  not  geo- 
metrical problems.    The  solution  of  these  problems  is  required 
from  Mechanics;  by  Geometry  the  use  of  them,  when  solved,  is 
shown.  .  .  .  Therefore   Geometry   is  founded   in  mechanical 
practice,  and  is  nothing  but  that  part  of  universal  Mechanics 
which  accurately  proposes  and  demonstrates  the  art  of  measur- 
ing.   But  since  the  manual  arts  are  chiefly  conversant  in  the 


312  MEMORABILIA   MATHEMATICA 

moving  of  bodies,  it  comes  to  pass  that  Geometry  is  commonly 
referred  to  their  magnitudes,  and  Mechanics  to  their  motion. 

NEWTON. 

Philosophiae    Naturalis    Principia    Mathe- 
matica,  Praefat. 

1866.  We  must,  then,  admit  .  .  .  that  there  is  an  independ- 
ent science  of  geometry  just  as  there  is  an  independent  science 
of  physics,  and  that  either  of  these  may  be  treated  by  mathe- 
matical methods.    Thus  geometry  becomes  the  simplest  of  the 
natural  sciences,  and  its  axioms  are  of  the  nature  of  physical 
laws,  to  be  tested  by  experience  and  to  be  regarded  as  true  only 
within  the  limits  of  error  of  observation — BOCHER,  MAXIME. 

Bulletin    American    Mathematical    Society, 
Vol.  2  (1904),  P-  124- 

1867.  Geometry  is  not  an  experimental  science;  experience 
forms  merely  the  occasion  for  our  reflecting  upon  the  geometri- 
cal ideas  which  pre-exist  in  us.    But  the  occasion  is  necessary, 
if  it  did  not  exist  we  should  not  reflect,  and  if  our  experiences 
were  different,  doubtless  our  reflections  would  also  be  different. 
Space  is  not  a  form  of  sensibility;  it  is  an  instrument  which 
serves  us  not  to  represent  things  to  ourselves,  but  to  reason  upon 
things. — POINCARE,  H. 

On  the  Foundations  of. Geometry;  Monist,  Vol.  9 
(1898-1899),  p.  41. 

1868.  It  has  been  said  that  geometry  is  an  instrument.    The 
comparison  may  be  admitted,  provided  it  is  granted  at  the 
same  time  that  this  instrument,  like  Proteus  in  the  fable,  ought 
constantly  to  change  its  form. — ARAGO. 

Oeuvres,  t.  2  (1854),  P-  694. 

1869.  It  is  essential  that  the  treatment  [of  geometry]  should 
be  rid  of  everything  superfluous,  for  the  superfluous  is  an  obsta- 
cle to  the  acquisition  of  knowledge;  it  should  select  everything 
that  embraces  the  subject  and  brings  it  to  a  focus,  for  this  is  of 
the  highest  service  to  science;  it  must  have  great  regard  both  to 
clearness  and  to  conciseness,  for  their  opposites  trouble  our 
understanding;  it  must  aim  to  generalize  its  theorems,  for  the 


GEOMETRY  313 

division  of  knowledge  into  small  elements  renders  it  difficult  of 
comprehension. — PROCLUS. 

Quoted  in  D.  E.  Smith:  The  Teaching  of  Geom- 
etry (Boston,  1911),  p.  71. 

1870.  Many  are  acquainted  with  mathematics,  but  mathesis 
few  know.    For  it  is  one  thing  to  know  a  number  of  propositions 
and  to  make  some  obvious  deductions  from  them,  by  accident 
rather  than  by  any  sure  method  of  procedure,  another  thing  to 
know  clearly  the  nature  and  character  of  the  science  itself,  to 
penetrate  into  its  inmost  recesses,  and  to  be  instructed  by  its 
universal  principles,  by  which  facility  in  working  out  countless 
problems  and  their  proofs  is  secured.    For  as  the  majority  of 
artists,  by  copying  the  same  model  again  and  again,  gain  certain 
technical  skill  in  painting,  but  no  other  knowledge  of  the  art  of 
painting  than  what  their  eyes  suggest,  so  many,  having  read 
the  books  of  Euclid  and  other  geometricians,  are  wont  to  devise, 
in  imitation  of  them  and  to  prove  some  propositions,  but  the 
most  profound  method  of  solving  more  difficult  demonstrations 
and  problems  they  are  utterly  ignorant  of. — LAFAILLE,  J.  C. 

Theoremata de  Centra  Gravitatis  (Anvers,  1632} , 
Praefat. 

1871.  The  elements  of  plane  geometry  should  precede  algebra 
for  every  reason  known  to  sound  educational  theory.     It  is 
more  fundamental,  more  concrete,  and  it  deals  with  things  and 
their  relations  rather  than  with  symbols. — BUTLER,  N.  M. 

The  Meaning  of  Education  etc.  (New  York, 
1905),  p.  171. 

1872.  The  reason  why  geometry  is  not  so  difficult  as  algebra, 
is  to  be  found  in  the  less  general  nature  of  the  symbols  employed. 
In  algebra  a  general  proposition  respecting  numbers  is  to  be 
proved.    Letters  are  taken  which  may  represent  any  of  the 
numbers  in  question,  and  the  course  of  the  demonstration,  far 
from  making  use  of  a  particular  case,  does  not  even  allow  that 
any  reasoning,  however  general  in  its  nature,  is  conclusive, 
unless  the  symbols  are  as  general  as  the  arguments.  ...  In 
geometry  on  the  contrary,  at  least  in  the  elementary  parts,  any 
proposition  may  be  safely  demonstrated  on  reasonings  on  any 
one  particular  example.  ...  It  also  affords  some  facility  that 


314  MEMORABILIA   MATHEMATICA 

the  results  of  elementary  geometry  are  in  many  cases  sufficiently 
evident  of  themselves  to  the  eye;  for  instance,  that  two  sides  of  a 
triangle  are  greater  than  the  third,  whereas  in  algebra  many 
rudimentary  propositions  derive  no  evidence  from  the  senses; 
for  example,  that  a3  -  b3  is  always  divisible  without  a  remainder 
by  a-b. — DE  MORGAN,  A. 

On  the  Study  and  Difficulties  of  Mathematics 

(Chicago,  1902],  chap.  13. 

1873.  The  principal  characteristics  of  the  ancient  geometry 
are: — 

(1)  A  wonderful  clearness  and  definiteness  of  its  concepts  and 
an  almost  perfect  logical  rigour  of  its  conclusions. 

(2)  A  complete  want  of  general  principles  and  methods.  .  .  . 
In  the  demonstration  of  a  theorem,  there  were,  for  the  ancient 
geometers,  as  many  different  cases  requiring  separate  proof  as 
there  were  different  positions  of  the  lines.    The  greatest  geome- 
ters considered  it  necessary  to  treat  all  possible  cases  independ- 
ently of  each  other,  and  to  prove  each  with  equal  fulness.    To 
devise  methods  by  which  all  the  various  cases  could  all  be 
disposed  of  with  one  stroke,  was  beyond  the  power  of  the 
ancients. — CAJORI,  F. 

History  of  Mathematics  (New  York,  1897), 
p.  62. 

1874.  It  has  been  observed  that  the  ancient  geometers  made 
use  of  a  kind  of  anaylsis,  which  they  employed  in  the  solution  of 
problems,  although  they  begrudged  to  posterity  the  knowledge  of 
it. — DESCARTES. 

Rules  for  the  Direction  of  the  Mind;  The  Philos- 
ophy of  Descartes  [Torrey]  (New  York,  1892), 
p.  68. 

1875.  The  ancients  studied  geometry  with  reference  to  the 
bodies  under  notice,  or  specially:  the  moderns  study  it  with 
reference  to  the  phenomena  to  be  considered,  or  generally.    The 
ancients  extracted  all  they  could  out  of  one  line  or  surface,  before 
passing  to  another;  and  each  inquiry  gave  little  or  no  assistance 
in  the  next.    The  moderns,  since  Descartes,  employ  themselves 
on  questions  which  relate  to  any  figure  whatever.    They  ab- 
stract, to  treat  by  itself,  every  question  relating  to  the  same 


GEOMETRY  315 

geometrical  phenomenon,  in  whatever  bodies  it  may  be  con- 
sidered. Geometers  can  thus  rise  to  the  study  of  new  geo- 
metrical conceptions,  which,  applied  to  the  curves  investigated 
by  the  ancients,  have  brought  out  new  properties  never  sus- 
pected by  them. — COMTE. 

Positive  Philosophy  [Martineau]  Bk.l,  chap.  3. 

1876.  It  is  astonishing  that  this  subject  [projective  geometry] 
should  be  so  generally  ignored,  for  mathematics  offers  nothing 
more  attractive.    It  possesses  the  concreteness  of  the  ancient 
geometry  without  the  tedious  particularity,  and  the  power  of 
the  analytical  geometry  without  the  reckoning,  and  by  the 
beauty  of  its  ideas  and  methods  illustrates  the  esthetic  generality 
which  is  the  charm  of  higher  mathematics,  but  which  the  ele- 
mentary mathematics  generally  lacks. 

Report  of  the  Committee  of  Ten  on  Secondary 
School  Studies  (Chicago,  1894),  P-  H6. 

1877.  There  exist  a  small  number  of  very  simple  fundamental 
relations  which  contain  the  scheme,  according  to  which  the 
remaining  mass  of  theorems  [in  projective  geometry]  permit  of 
orderly  and  easy  development. 

By  a  proper  appropriation  of  a  few  fundamental  relations  one 
becomes  master  of  the  whole  subject;  order  takes  the  place  of 
chaos,  one  beholds  how  all  parts  fit  naturally  into  each  other, 
and  arrange  themselves  serially  in  the  most  beautiful  order,  and 
how  related  parts  combine  into  well-defined  groups.  In  this 
manner  one  arrives,  as  it  were,  at  the  elements,  which  nature 
herself  employs  in  order  to  endow  figures  with  numberless 
properties  with  the  utmost  economy  and  simplicity. 

STEINER,  J. 
Werke,  Bd.  1   (1881),  p.  288. 

1878.  Euclid  once  said  to  his  king  Ptolemy,  who,  as  is  easily 
understood,  found  the  painstaking  study  of  the  "Elements" 
repellant,  "There  exists  no  royal  road  to  mathematics."    But 
we  may  add:  Modern  geometry  is  a  royal  road.    It  has  dis- 
closed "the  organism,  by  means  of  which  the  most  heterogeneous 
phenomena  in  the  world  of  space  are  united  one  with  another" 


316  MEMORABILIA   MATHEMATICA 

(Steiner),  and  has,  as  we  may  say  without  exaggeration,  almost 
attained  to  the  scientific  ideal. — HANKEL,  H. 

Die   Entwickelung   der   Mathematik   in   den 
letzten  Jahrhunderten  (Tubingen,  1869). 

1879.  The  two  mathematically  fundamental  things  in  pro- 
jective  geometry  are  anharmonic  ratio,  and  the  quadrilateral 
construction.     Everything  else  follows  mathematically  from 
these  two. — RUSSELL,  BERTRAND. 

Foundations  of  Geometry  (Cambridge,  1897), 
p.  122. 

1880.  .  .  .  Projective   Geometry:    a   boundless   domain   of 
countless  fields  where  reals  and  imaginaries,  finites  and  in- 
finites, enter  on  equal  terms,  where  the  spirit  delights  in  the 
artistic  balance  and  symmetric  interplay  of  a  kind  of  con- 
ceptual and  logical  counterpoint, — an  enchanted  realm  where 
thought  is  double  and  flows  throughout  hi  parallel  streams. 

KEYSER,  C.  J. 

Lectures   on   Science,    Philosophy   and   Arts 
(New  York,  1908),  p.  2. 

1881.  The  ancients,  hi  the  early  days  of  the  science,  made 
great  use  of  the  graphic  method,  even  hi  the  form  of  construc- 
tion; as  when  Aristarchus  of  Samos  estimated  the  distance  of 
the  sun  and  moon  from  the  earth  on  a  triangle  constructed  as 
nearly  as  possible  in  resemblance  to  the  right-angled  triangle 
formed  by  the  three  bodies  at  the  instant  when  the  moon  is  in 
quadrature,  and  when  therefore  an  observation  of  the  angle  at 
the   earth  would   define   the   triangle.     Archimedes   himself, 
though  he  was  the  first  to  introduce  calculated  determinations 
into  geometry,  frequently  used  the  same  means.    The  introduc- 
tion of  trigonometry  lessened  the  practice;  but  did  not  abolish 
it.    The  Greeks  and  Arabians  employed  it  still  for  a  great  num- 
ber of  investigations  for  which  we  now  consider  the  use  of  the 
Calculus  indispensable. — COMTE,  A. 

Positive  Philosophy  [Martineau],  Bk.l,  chap.  3. 

1882.  A  mathematical  problem  may  usually  be  attacked  by 
what  is  termed  in  military  parlance  the  method  of  "systematic 
approach;"  that  is  to  say,  its  solution  may  be  gradually  felt 


GEOMETRY  317 

for,  even  though  the  successive  steps  leading  to  that  solution 
cannot  be  clearly  foreseen.  But  a  Descriptive  Geometry 
problem  must  be  seen  through  and  through  before  it  can  be 
attempted.  The  entire  scope  of  its  conditions,  as  well  as  each 
step  toward  its  solution,  must  be  grasped  by  the  imagination. 
It  must  be  "taken  by  assault." — CLARKE,  G.  S. 

Quoted  in  W.  S.  Hall:  Descriptive  Geometry 

(New  York,  1902),  chap.  1. 

1883.  The  grand  use  [of  Descriptive  Geometry]  is  in  its  appli- 
cation to  the  industrial  arts; — its  few  abstract  problems,  capable 
of  invariable  solution,  relating  essentially  to  the  contacts  and 
intersections  of  surfaces;  so  that  all  the  geometrical  questions 
which  may  arise  in  any  of  the  various  arts  of  construction, — as 
stone-cutting,  carpentry,  perspective,  dialing,  fortification, 
etc., — can  always  be  treated  as  simple  individual  cases  of  a 
single  theory,  the  solution  being  certainly  obtainable  through 
the  particular  circumstances  of  each  case.  This  creation  must 
be  very  important  in  the  eyes  of  philosophers  who  think  that  all 
human  achievement,  thus  far,  is  only  a  first  step  toward  a 
philosophical  renovation  of  the  labours  of  mankind;  towards 
that  precision  and  logical  character  which  can  alone  ensure  the 
future  progression  of  all  arts.  ...  Of  Descriptive  Geometry, 
it  may  further  be  said  that  it  usefully  exercises  the  student's 
faculty  of  Imagination, — of  conceiving  of  complicated  geo- 
metrical combinations  in  space;  and  that,  while  it  belongs  to  the 
geometry  of  the  ancients  by  the  character  of  its  solutions,  it 
approaches  to  the  geometry  of  the  moderns  by  the  nature  of  the 
questions  which  compose  it. — COMTE,  A. 

Positive  Philosophy  [Martineau]  Bk.  1,  chap.  8. 

1884.  There  is  perhaps  nothing  which  so  occupies,  as  it  were, 
the  middle  position  of  mathematics,  as  trigonometry. 

HERBART,  J.  F. 

Idee   eines   ABC   der   Anschauung;    Werke 
(Kehrbach)  (Langensalza,  1890),  Bd.  1,  p.  174- 

1885.  Trigonometry    contains    the    science    of    continually 
undulating   magnitude:   meaning   magnitude   which   becomes 
alternately  greater  and  less,  without  any  termination  to  succes- 


318  MEMORABILIA   MATHEMATICA 

sion  of  increase  and  decrease.  .  .  .  All  trigonometric  functions 
are  not  undulating:  but  it  may  be  stated  that  in  common  algebra 
nothing  but  infinite  series  undulate :  in  trigonometry  nothing  but 
infinite  series  do  not  undulate. — DE  MORGAN,  A. 

Trigonometry  and  Double  Algebra  (London, 
1849),  Bk.l,  chap.  1. 

1886.  Sin2<£  is  odious  to  me,  even  though  Laplace  made  use  of 
it;  should  it  be  feared  that  sin</>2  might  become  ambiguous, 
which  would  perhaps  never  occur,  or  at  most  very  rarely  when 
speaking  of  sin  (</>2),  well  then,  let  us  write  (sin</>)2,  but  not  sin2<£, 
which  by  analogy  should  signify  sin  (sin<£). — GAUSS. 

Gauss-Schumacher  Briefwechsel,  Bd.  8,  p.  292; 
Bd.  4,  p.  63. 

1887.  Perhaps  to  the  student  there  is  no  part  of  elementary 
mathematics  so  repulsive  as  is  spherical  trigonometry. 

TAIT,  P.  G. 

Encyclopedia  Britannica,  9th  Edition;  Article 
"Quaternions." 

1888.  "Napier's  Rule  of  circular  parts"  is  perhaps  the  hap- 
piest example  of  artificial  memory  that  is  known. — CAJORI,  F. 

History  of  Mathematics  (New   York,  1897), 
p.  165. 

1889.  The  analytical  equations,  unknown  to  the  ancients, 
which  Descartes  first  introduced  into  the  study  of  curves  and 
surfaces,  are  not  restricted  to  the  properties  of  figures,  and  to 
those  properties  which  are  the  object  of  rational  mechanics; 
they  apply  to  all  phenomena  in  general.    There  cannot  be  a 
language  more  universal  and  more  simple,  more  free  from  errors 
and  obscurities,  that  is  to  say,  better  adapted  to  express  the 
invariable  relations  of  nature. — FOURIER. 

Theorie  Analytique  de  la  Chaleur,  Discours 
Preliminaire. 

1890.  It  is  impossible  not  to  feel  stirred  at  the  thought  of  the 
emotions  of  men  at  certain  historic  moments  of  adventure  and 
discovery — Columbus  when  he  first  saw  the  Western  shore, 
Pizarro  when  he  stared  at  the  Pacific  Ocean,  Franklin  when  the 


GEOMETRY  319 

electric  spark  came  from  the  string  of  his  kite,  Galileo  when  he 
first  turned  his  telescope  to  the  heavens.  Such  moments  are 
also  granted  to  students  in  the  abstract  regions  of  thought,  and 
high  among  them  must  be  placed  the  morning  when  Descartes 
lay  in  bed  and  invented  the  method  of  co-ordinate  geometry. 

WHITEHEAD,  A.  N. 

An  Introduction  to  Mathematics  (New  York, 

1911),  p.  122. 

1891.  It  is  often  said  that  an  equation  contains  only  what  has 
been  put  into  it.    It  is  easy  to  reply  that  the  new  form  under 
which  things  are  found  often  constitutes  by  itself  an  important 
discovery.     But  there  is  something  more:  analysis,  by  the 
simple  play  of  its  symbols,  may  suggest  generalizations  far 
beyond  the  original  limits. — PICARD,  E. 

Bulletin    American    Mathematical    Society, 
Vol.  2  (1905),  p.  409. 

1892.  It  is  not  the  Simplicity  of  the  Equation,  but  the 
Easiness  of  the  Description,  which  is  to  determine  the  Choice  of 
our  Lines  for  the  Constructions  of  Problems.    For  the  Equation 
that  expresses  a  Parabola  is  more  simple  than  that  that  expresses 
the  Circle,  and  yet  the  Circle,  by  its  more  simple  Construction, 

is  admitted  before  it. — NEWTON. 

The  Linear  Constructions  of  Equations; 
Universal  Arithmetic  (London,  1769),  Vol.  2, 
p.  468. 

1893.  The  pursuit   of  mathematics   unfolds   its   formative 
power  completely  only  with  the  transition  from  the  elementary 
subjects  to  analytical  geometry.    Unquestionably  the  simplest 
geometry  and  algebra  already  accustom  the  mind  to  sharp 
quantitative  thinking,  as  also  to  assume  as  true  only  axioms  and 
what  has  been  proven.    But  the  representation  of  functions  by 
curves  or  surfaces  reveals  a  new  world  of  concepts  and  teaches 
the  use  of  one  of  the  most  fruitful  methods,  which  the  human 
mind  ever  employed  to  increase  its  own  effectiveness.    What  the 
discovery  of  this  method  by  Vieta  and  Descartes  brought  to 
humanity,  that  it  brings  today  to  every  one  who  is  in  any 
measure  endowed  for  such  things:  a  life-epoch-making  beam  of 
light  [Lichtblick].    This  method  has  its  roots  in  the  farthest 


320  MEMORABILIA    MATHEMATICA 

depths  of  human  cognition  and  so  has  an  entirely  different 
significance,  than  the  most  ingenious  artifice  which  serves  a 
special  purpose. — BOIS-REYMOND,  EMIL  DU. 

Reden,  Bd.  1  (Leipzig,  1885),  p.  287. 

1894.  Song  of  the  Screw. 

A  moving  form  or  rigid  mass, 

Under  whate'er  conditions 
Along  successive  screws  must  pass 

Between  each  two  positions. 
It  turns  around  and  slides  along — 
This  is  the  burden  of  my  song. 

The  pitch  of  screw,  if  multiplied 

By  angle  of  rotation, 
Will  give  the  distance  it  must  glide 

In  motion  of  translation. 
Infinite  pitch  means  pure  translation, 
And  zero  pitch  means  pure  rotation. 

Two  motions  on  two  given  screws, 

With  amplitudes  at  pleasure, 
Into  a  third  screw-motion  fuse, 

Whose  amplitude  we  measure 
By  parallelogram  construction 
(A  very  obvious  deduction) . 

Its  axis  cuts  the  nodal  line 

Which  to  both  screws  is  normal, 

And  generates  a  form  divine, 

Whose  name,  in  language  formal, 

Is  "surface-ruled  of  third  degree." 

Cylindroid  is  the  name  for  me. 

Rotation  round  a  given  line 

Is  like  a  force  along, 
If  to  say  couple  you  decline, 

You're  clearly  in  the  wrong; — 
'Tis  obvious,  upon  reflection, 
A  line  is  not  a  mere  direction. 


GEOMETRY  321 

So  couples  with  translations  too 

In  all  respects  agree; 
And  thus  there  centres  in  the  screw 

A  wondrous  harmony 
Of  Kinematics  and  of  Statics, — 
The  sweetest  thing  in  mathematics. 

The  forces  on  one  given  screw, 

With  motion  on  a  second, 
In  general  some  work  will  do, 

Whose  magnitude  is  reckoned 
By  angle,  force,  and  what  we  call 
The  coefficient  virtual. 

Rotation  now  to  force  convert, 

And  force  into  rotation; 
Unchanged  the  work,  we  can  assert, 

In  spite  of  transformation. 
And  if  two  screws  no  work  can  claim, 
Reciprocal  will  be  their  name. 

Five  numbers  will  a  screw  define, 

A  screwing  motion,  six; 
For  four  will  give  the  axial  line, 

One  more  the  pitch  will  fix; 
And  hence  we  always  can  contrive 
One  screw  reciprocal  to  five. 

Screws — two,  three,  four  or  five,  combined 

(No  question  here  of  six), 
Yield  other  screws  which  are  confined 

Within  one  screw  complex. 
Thus  we  obtain  the  clearest  notion 
Of  freedom  and  constraint  of  motion. 

In  complex  III,  three  several  screws 

At  every  point  you  find, 
Or  if  you  one  direction  choose, 

One  screw  is  to  your  mind; 


322  MEMORABILIA   MATHEMATICA 

And  complexes  of  order  III. 
Their  own  reciprocals  may  be. 

In  IV,  wherever  you  arrive, 

You  find  of  screws  a  cone, 
On  every  line  of  complex  V. 

There  is  precisely  one; 
At  each  point  of  this  complex  rich, 
A  plane  of  screws  have  given  pitch. 

But  time  would  fail  me  to  discourse 

Of  Order  and  Degree; 
Of  Impulse,  Energy  and  Force, 

And  Reciprocity. 

All  these  and  more,  for  motions  small, 
Have  been  discussed  by  Dr.  Ball. 

ANONYMOUS. 


CHAPTER  XIX 

THE  CALCULUS  AND  ALLIED  TOPICS 

1901.  It  may  be  said  that  the  conceptions  of  differential 
quotient  and  integral,  which  in  their  origin  certainly  go  back  to 
Archimedes,  were  introduced  into  science  by  the  investigations 
of  Kepler,  Descartes,  Cavalieri,  Fermat  and  Wallis.  .  .  .  The 
capital    discovery    that    differentiation    and    integration    are 
inverse  operations  belongs  to  Newton  and  Leibnitz. 

LIE,  SOPHUS. 

Leipziger  Berichte,   47   (1895),   Math.-phys. 
Classe,  p.  53. 

1902.  It   appears  that  Fermat,   the  true  inventor  of  the 
differential  calculus,  considered  that  calculus  as  derived  from 
the  calculus  of  finite  differences  by  neglecting  infinitesimals  of 
higher  orders  as  compared  with  those  of  a  lower  order.  .  .  . 
Newton,  through  his  method  of  fluxions,  has  since  rendered  the 
calculus  more  analytical,  he  also  simplified  and  generalized  the 
method  by  the  invention  of  his  binomial  theorem.     Leibnitz 
has  enriched  the  differential  calculus  by  a  very  happy  notation. 

LAPLACE. 

Les  Integrates  Definies,  etc.;  Oeuvres,  t.  12 
(Paris,  1898),  p.  359. 

1903.  Professor  Peacock's  Algebra,  and  Mr.  Whewell's  Doc- 
trine of  Limits  should  be  studied  by  every  one  who  desires  to 
comprehend  the  evidence  of  mathematical  truths,   and  the 
meaning  of  the  obscure  processes  of  the  calculus;  while,  even 
after  mastering  these  treatises,  the  student  will  have  much  to 
learn  on  the  subject  from  M.  Comte,  of  whose  admirable  work 
one  of  the  most  admirable  portions  is  that  in  which  he  may 
truly  be  said  to  have  created  the  philosophy  of  the  higher 
mathematics. — MILL,  J.  S. 

System  of  Logic,  Bk.  3,  chap.  24,  sect.  6. 
323 


324  MEMORABILIA  MATHEMATICA 

1904.  If  we  must  confine  ourselves  to  one  system  of  notation 
then  there  can  be  no  doubt  that  that  which  was  invented  by 
Leibnitz  is  better  fitted  for  most  of  the  purposes  to  which  the 
infinitesimal  calculus  is  applied  than  that  of  fluxions,  and  for 
some  (such  as  the  calculus  of  variations)  it  is  indeed  almost 
essential.— BALL,  W.  W.  R. 

History    of    Mathematics    (London,    1901), 
p.  371. 

1905.  The  difference  between  the  method  of  infinitesimals 
and  that  of  limits  (when  exclusively  adopted)  is,  that  in  the 
latter  it  is  usual  to  retain  evanescent  quantities  of  higher  orders 
until  the  end  of  the  calculation  and  then  neglect  them.    On  the 
other  hand,  such  quantities  are  neglected  from  the  commence- 
ment in  the  infinitesimal  method,  from  the  conviction  that 
they  cannot  affect  the  final  result,  as  they  must  disappear  when 
we  proceed  to  the  limit. — WILLIAMSON,  B. 

Encyclopedia  Britannica,  9th  Edition;  Article 
"Infinitesimal  Calculus,"  sect.  14- 

1906.  When  we  have  grasped  the  spirit  of  the  infinitesimal 
method,  and  have  verified  the  exactness  of  its  results  either  by 
the  geometrical  method  of  prime  and  ultimate  ratios,  or  by  the 
analytical  method  of  derived  functions,  we  may  employ  in- 
finitely small  quantities  as  a  sure  and  valuable  means  of  shorten- 
ing and  simplifying  our  proofs. — LAGKANGE. 

Mechanique    Analytique,    Preface;    Oeuvres, 
t.  2  (Paris,  1888),  p.  14. 

1907.  The  essential  merit,  the  sublimity,  of  the  infinitesimal 
method  lies  in  the  fact  that  it  is  as  easily  performed  as  the 
simplest  method  of  approximation,  and  that  it  is  as  accurate 
as  the  results  of  an  ordinary  calculation.    This  advantage  would 
be  lost,  or  at  least  greatly  impaired,  if,  under  the  pretense  of 
securing  greater  accuracy  throughout  the  whole  process,  we 
were  to  substitute  for  the  simpler  method  given  by  Leibnitz, 
one  less  convenient  and  less  in  harmony  with  the  probable 
course  of  natural  events.  .  .  . 

The  objections  which  have  been  raised  against  the  infinites- 
imal method  are  based  on  the  false  supposition  that  the  errors 


THE   CALCULUS   AND   ALLIED  TOPICS  325 

due  to  neglecting  infinitely  small  quantities  during  the  actual 
calculation  will  continue  to  exist  in  the  result  of  the  calculation. 

CARNOT,  L. 

Reflections   sur  la   Metaphysique  du   Calcul 
Infinitesimal  (Paris,  1818),  p.  215. 

1908.  A  limiting  ratio  is  neither  more  nor  less  difficult  to 
define  than  an  infinitely  small  quantity. — CARNOT,  L. 

Reflections  sur  la  Metaphysique  du  Calcul 
Infinitesimal  (Paris,  1818),  p.  210. 

1909.  A  limit  is  a  peculiar  and  fundamental  conception,  the 
use  of  which  in  proving  the  propositions  of  Higher  Geometry 
cannot  be  superseded  by  any  combination  of  other  hypotheses 
and  definitions.    The  axiom  just  noted  that  what  is  true  up  to 
the  limit  is  true  at  the  limit,  is  involved  in  the  very  conception 
of  a  limit:  and  this  principle,  with  its  consequences,  leads  to  all 
the  results  which  form  the  subject  of  the  higher  mathematics, 
whether  proved  by  the  consideration  of  evanescent  triangles, 
by  the  processes  of  the  Differential  Calculus,  or  in  any  other 

way. — WHEWELL,  W. 

The   Philosophy   of  the   Inductive   Sciences, 
Part  1,  bk.  2,  chap.  12,  sect.  1,  (London,  1858). 

1910.  The  differential  calculus  has  all  the  exactitude  of  other 

algebraic  operations. — LAPLACE. 

Theorie  Analytique  des  Probabilites,  Introduc- 
tion; Oeuvres,  t.  7  (Paris,  1886),  p.  S7. 

1911.  The  method  of  fluxions  is  probably  one  of  the  greatest, 
most  subtle,  and  sublime  discoveries  of  any  age:  it  opens  a  new 
world  to  our  view,  and  extends  our  knowledge,  as  it  were,  to 
infinity;  carrying  us  beyond  the  bounds  that  seemed  to  have 
been  prescribed  to  the  human  mind,  at  least  infinitely  beyond 
those  to  which  the  ancient  geometry  was  confined. 

HTJTTON,  CHARLES. 

A  Philosophical  and  Mathematical  Dictionary 
(London,  1815),  Vol.  1,  p.  525. 

1912.  The  states  and  conditions  of  matter,  as  they  occur  in 
nature,  are  in  a  state  of  perpetual  flux,  and  these  qualities  may 


326  MEMORABILIA   MATHEMATICA 

be  effectively  studied  by  the  Newtonian  method  (Methodus 
fluxionem)  whenever  they  can  be  referred  to  number  or  sub- 
jected to  measurement  (real  or  imaginary).  By  the  aid  of 
Newton's  calculus  the  mode  of  action  of  natural  changes  from 
moment  to  moment  can  be  portrayed  as  faithfully  as  these 
words  represent  the  thoughts  at  present  in  my  mind.  From 
this,  the  law  which  controls  the  whole  process  can  be  determined 
with  unmistakable  certainty  by  pure  calculation. 

MELLOR,  J.  W. 

Higher  Mathematics  for  Students  of  Chemistry 
and  Physics  (London,  1902),  Prologue. 

1913.  The  calculus  is  the  greatest  aid  we  have  to  the  ap- 
preciation of  physical  truth  in  the  broadest  sense  of  the  word. 

OSGOOD,  W.  F. 

Bulletin    American    Mathematical    Society, 
Vol.  13  (1907),  p.  467. 

1914.  [Infinitesimal]  analysis  is  the  most  powerful  weapon  of 
thought  yet  devised  by  the  wit  of  man. — SMITH,  W.  B. 

Infinitesimal   Analysis    (New    York,    1898), 
Preface,  p.  vii. 

1915.  The  method  of  Fluxions  is  the  general  key  by  help 
whereof  the  modern  mathematicians  unlock  the  secrets  of 
Geometry,  and  consequently  of  Nature.    And,  as  it  is  that 
which  hath  enabled  them  so  remarkably  to  outgo  the  ancients  hi 
discovering  theorems  and  solving  problems,  the  exercise  and 
application  thereof  is  become  the  main  if  not  sole  employment  of 
all  those  who  in  this  age  pass  for  profound  geometers. 

BERKELEY,  GEORGE. 

The  Analyst,  sect.  3. 

1916.  I  have  at  last  become  fully  satisfied  that  the  language 
and  idea  of  infinitesimals  should  be  used  in  the  most  elementary 
instruction — under  all  safeguards  of  course. — DE  MORGAN,  A. 

Graves'  Life  of  W.  R.  Hamilton  (New  York, 
1882-1889),  Vol.  3,  p.  479. 

1917.  Pupils  should  be  taught  how  to  differentiate  and  how  to 
integrate  simple  algebraic  expressions  before  we  attempt  to 


THE  CALCULUS  AND  ALLIED  TOPICS          327 

teach  them  geometry  and  these  other  complicated  things.  The 
dreadful  fear  of  the  symbols  is  entirely  broken  down  in  those 
cases  where  at  the  beginning  the  teaching  of  the  calculus  is 
adopted.  Then  after  the  pupil  has  mastered  those  symbols  you 
may  begin  geometry  or  anything  you  please.  I  would  also 
abolish  out  of  the  school  that  thing  called  geometrical  conies. 
There  is  a  great  deal  of  superstition  about  conic  sections.  The 
student  should  be  taught  the  symbols  of  the  calculus  and  the 
simplest  use  of  these  symbols  at  the  earliest  age,  instead  of 
these  being  left  over  until  he  has  gone  to  the  College  or  Univer- 
sity.— THOMPSON,  S.  P. 

Perry's   Teaching  of  Mathematics   (London, 

1902),  p.  49. 

1918.  Every  one  versed  in  the  matter  will  agree  that  even  the 
elements  of  a  scientific  study  of  nature  can  be  understood  only 
by  those  who  have  a  knowledge  of  at  least  the  elements  of  the 
differential  and  integral  calculus,  as  well  as  of  analytical  ge- 
ometry— i.  e.  the  so-called  lower  part  of  the  higher  mathe- 
matics. .  .  .  We  should  raise  the  question,  whether  sufficient 
time  could  not  be  reserved  in  the  curricula  of  at  least  the  science 
high  schools  [Realanstalten]  to  make  room  for  these  sub- 
jects. .  .  . 

The  first  consideration  would  be  to  entirely  relieve  from  the 
mathematical  requirements  of  the  university  [Hochschule] 
certain  classes  of  students  who  can  get  along  without  extended 
mathematical  knowledge,  or  to  make  the  necessary  mathe- 
matical knowledge  accessible  to  them  in  a  manner  which,  for 
various  reasons,  has  not  yet  been  adopted  by  the  university. 
Among  such  students  I  would  count  architects,  also  the  chemists 
and  in  general  the  students  of  the  so-called  descriptive  natural 
sciences.  I  am  moreover  of  the  opinion — and  this  has  been  for 
long  a  favorite  idea  of  mine — ,  that  it  would  be  very  useful  to 
medical  students  to  acquire  such  mathematical  knowledge  as  is 
indicated  by  the  above  described  modest  limits;  for  it  seems  im- 
possible to  understand  far-reaching  physiological  investigations, 
if  one  is  terrified  as  soon  as  a  differential  or  integration  symbol 

appears. — KLEIN,  F. 

Jahresbericht    der    Deutschen    Mathematiker 
Vereinigung,  Bd.  2  (1902),  p.  181, 


328  MEMORABILIA   MATHEMATICA 

1919.  Common  integration  is  only  the  memory  of  differentia- 
tion .  .  .  the  different  artifices  by  which  integration  is  effected, 
are  changes,  not  from  the  known  to  the  unknown,  but  from 
forms  in  which  memory  will  not  serve  us  to  those  in  which  it  will. 

DE  MORGAN,  A. 

Transactions  Cambridge  Philosophical  Society, 
Vol.  8  (1844),  P.  188. 

1920.  Given  for  one  instant  an  intelligence  which  could 
comprehend  all  the  forces  by  which  nature  is  animated  and  the 
respective  positions  of  the  beings  which  compose  it,  if  moreover 
this  intelligence  were  vast  enough  to  submit  these  data  to 
analysis,  it  would  embrace  in  the  same  formula  both  the  move- 
ments of  the  largest  bodies  in  the  universe  and  those  of  the 
lightest  atom :  to  it  nothing  would  be  uncertain,  and  the  future 
as  the  past  would  be  present  to  its  eyes.    The  human  mind 
offers  a  feeble  outline  of  that  intelligence,  hi  the  perfection  which 
it  has  given  to  astronomy.    Its  discoveries  hi  mechanics  and  in 
geometry,  joined  to  that  of  universal  gravity,  have  enabled  it  to 
comprehend  in  the  same  analytical  expressions  the  past  and 
future  states  of  the  world  system. — LAPLACE. 

Theorie  Analytique  des  Probabilites,  Introduc- 
tion; Oeuvres,  t.  7  (Paris,  1886),  p.  6. 

1921.  There  is  perhaps  the  same  relation  between  the  action 
of  natural  selection  during  one  generation  and  the  accumulated 
result  of  a  hundred  thousand  generations,  that  there  exists 
between  differential  and  integral.    How  seldom  are  we  able  to 
follow  completely  this  latter  relation  although  we  subject  it  to 
calculation.    Do  we  on  that  account  doubt  the  correctness  of 
our  integrations? — BOIS-REYMOND,  EMIL  DU. 

Reden,  Bd.  1  (Leipzig,  1885),  p.  228. 

1922.  It  seems  to  be  expected  of  every  pilgrim  up  the  slopes 
of  the  mathematical  Parnassus,  that  he  will  at  some  point  or 
other  of  his  journey  sit  down  and  invent  a  definite  integral  or 
two  towards  the  increase  of  the  common  stock. 

STLVESTEB,  J.  J. 

Notes  to  the  Meditation  on  Poncelet's  Theorem; 
Mathematical  Papers,  Vol.  2,  p.  214- 


THE   CALCULUS  AND   ALLIED  TOPICS  329 

1923.  The  experimental  verification  of  a  theory  concerning 
any  natural  phenomenon  generally  rests  on  the  result  of  an 
integration. — MELLOR,  J.  W. 

Higher  Mathematics  for  Students  of  Chemistry 
and  Physics  (New  York,  1902} ,  p.  150. 

1924.  Among  all  the  mathematical  disciplines  the  theory  of 
differential  equations  is  the  most  important.  ...  It  furnishes 
the  explanation  of   all   those   elementary   manifestations   of 
nature  which  involve  time.  .  .  . — LIE,  SOPHUS. 

Leipziger  Berichte,   J+7    (1895);   Math.-^phys. 
Classe,  p.  262. 

1925.  If  the  mathematical  expression  of  our  ideas  leads  to 
equations  which  cannot  be  integrated,  the  working  hypothesis 
will  either  have  to  be  verified  some  other  way,  or  else  relegated 
to  the  great  repository  of  unverified  speculations. 

MELLOR,  J.  W. 

Higher  Mathematics  for  Students  of  Chemistry 
and  Physics  (New  York,  1902),  p.  157. 

1926.  It  is  well  known  that  the  central  problem  of  the  whole 
of  modern  mathematics  is  the  study  of  the  transcendental 
functions  defined  by  differential  equations. — KLEIN,  F. 

Lectures  on  Mathematics  (New  York,  1911), 
p.  8. 

1927.  Every  one  knows  what  a  curve  is,  until  he  has  studied 
enough  mathematics  to  become  confused  through  the  countless 
number  of  possible  exceptions.  ...  A  curve  is  the  totality  of 
points,  whose  co-ordinates  are  functions  of  a  parameter  which 
may  be  differentiated  as  often  as  may  be  required. — KLEIN,  F. 

Elementar  Mathematik  vom  hoheren  Stand- 
punkte  aus.  (Leipzig.  1909)  Vol.  2,  p.  854- 

1928.  Fourier's  theorem  is  not  only  one  of  the  most  beautiful 
results  of  modern  analysis,  but  it  may  be  said  to  furnish  an 
indispensable  instrument  in  the  treatment  of  nearly  every 
recondite   question   in   modern   physics.     To   mention   only 
sonorous  vibrations,  the  propagation  of  electric  signals  along 
telegraph  wires,  and  the  conduction  of  heat  by  the  earth's 


330  MEMORABILIA   MATHEMATICA 

crust,  as  subjects  in  their  generality  intractable  without  it,  is  to 
give  but  a  feeble  idea  of  its  importance. — THOMSON  AND  TAIT. 
Ekments  of  Natural  Philosophy,  chap.  1. 

1929.  The  principal  advantage  arising  from  the  use  of  hyper- 
bolic functions  is  that  they  bring  to  light  some  curious  analogies 
between  the  integrals  of  certain  irrational  functions. 

BYERLY,  W.  E. 

Integral  Calculus  (Boston,  1890),  p.  80. 

1930.  Hyperbolic  functions  are  extremely  useful  in  every 
branch  of  pure  physics  and  in  the  applications  of  physics  whether 
to  observational  and  experimental  sciences  or  to  technology. 
Thus  whenever  an  entity  (such  as  light,  velocity,  electricity,  or 
radio-activity)  is  subject  to  gradual  absorption  or  extinction,  the 
decay  is  represented  by  some  form  of  hyperbolic  functions. 
Mercator's    projection    is   likewise    computed    by    hyperbolic 
functions.     Whenever  mechanical  strains  are  regarded  great 
enough  to  be  measured  they  are  most  simply  expressed  in  terms 
of  hyperbolic  functions.     Hence  geological  deformations  in- 
variably lead  to  such  expressions.  .  .  . — WALCOTT,  C.  D. 

Smithsonian  Mathematical  Tables,  Hyberbolic 
Functions  (Washington,  1909),  Advertisement. 

1931.  Geometry  may  sometimes  appear  to  take  the  lead  over 
analysis,  but  in  fact  precedes  it  only  as  a  servant  goes  before  his 
master  to  clear  the  path  and  light  him  on  the  way.    The  interval 
between  the  two  is  as  wide  as  between  empiricism  and  science,  as 
between  the  understanding  and  the  reason,  or  as  between  the 
finite  and  the  infinite. — SYLVESTER,  J.  J. 

Philosophic  Magazine,  Vol.  81  (1866),  p.  521. 

1932.  Nature  herself  exhibits  to  us  measurable  and  observable 
quantities  in  definite  mathematical  dependence;  the  conception 
of  a  function  is  suggested  by  all  the  processes  of  nature  where  we 
observe  natural  phenomena  varying  according  to  distance  or  to 
time.    Nearly  all  the  "known"  functions  have  presented  them- 
selves in  the  attempt  to  solve  geometrical,  mechanical,   or 
physical  problems. — MERZ,  J.  T. 

A  History  of  European  Thought  in  the  Nine- 
teenth Century  (Edinburgh  and  London,  1903}, 
p.  696. 


THE  CALCULUS  AND  ALLIED  TOPICS          331 

1933.  That  flower  of  modern  mathematical  thought — the 
notion  of  a  function. — McCoRMACK,  THOMAS  J. 

On  the  Nature  of  Scientific  Law  and  Scientific 
Explanation,  Monist,  Vol.  10  (1899-1900), 
p.  555. 

1934.  Fuchs.  Ich  bin  von  alledem  so  consterniert, 

Als  wiirde  mir  ein  Kreis  im  Kopfe  quadriert. 

Meph.  Nachher  vor  alien  andern  Sachen 

Miisst   ihe  euch   an  die   Funktionen-Theorie 

machen. 

Da  seht,  dass  ihr  tiefsinnig  fasst, 
Was  sich  zu  integrieren  nicht  passt. 
An  Theoremen  wird's  euch  nicht  fehlen, 
Miisst  nur  die  Verschwindungspunkte  zahlen, 
Umkehren,  abbilden,  auf  der  Eb'ne  'rumfahren 
Und  mit  den  Theta-Produkten  nicht  sparen. 

LASSWITZ,  KURD. 

Der  Faust-Tragodie  (-n)ter  Tiel;  Zeitschrift 
fur  den  math.-natur.Unterricht,  Bd.14  (1883), 
p.  316. 

Fuchs.  Your  words  fill  me  with  an  awful  dread, 

Seems  like  a  circle  were  squared  in  my  head. 

Meph.  Next  in  order  you  certainly  ought 

On  function-theory  bestow  your  thought, 

And  penetrate  with  contemplation 

What  resists  your  attempts  at  integration. 

You'll  find  no  dearth  of  theorems  there — 

To  vanishing-points  give  proper  care — 

Enumerate,  reciprocate, 

Nor  forget  to  delineate, 

Traverse  the  plane  from  end  to  end, 

And  theta-f unctions  freely  spend. 

1935.  The  student  should  avoid  founding  results  upon  diver- 
gent series,  as  the  question  of  their  legitimacy  is  disputed  upon 
grounds  to  which  no  answer  commanding  anything  like  general 
assent  has  yet  been  given.  But  they  may  be  used  as  means  of 


332  MEMORABILIA   MATHEMATICA 

discovery,  provided  that  their  results  be  verified  by  other 
means  before  they  are  considered  as  established. 

DE  MORGAN,  A. 

Trigonometry  and  Doubk  Algebra   (London, 
1849),  p.  55. 

1936.  There  is  nothing  now  which  ever  gives  me  any  thought 
or  care  in  algebra  except  divergent  series,  which  I  cannot  follow 
the  French  in  rejecting. — DE  MORGAN,  A. 

Graves'  Life  of  W.  R.  Hamilton  (New  York, 
1882-1889),  Vol.  3,  p.  249. 

1937.  It  is  a  strange  vicissitude  of  our  science  that  these 
[divergent]  series  which  early  in  the  century  were  supposed  to  be 
banished  once  and  for  all  from  rigorous  mathematics  should  at 
its  close  be  knocking  at  the  door  for  readmission. — PIERPONT,  J. 

Congress  of  Arts  and  Sciences  (Boston  and 
New  York,  1905),  Vol.  1,  p.  476. 

1938.  Zeno  was  concerned  with  three  problems.  .  .  .  These 
are  the  problem  of  the  infinitesimal,  the  infinite,  and  con- 
tinuity. .  .  .  From  him  to  our  own  day,  the  finest  intellects  of 
each  generation  hi  turn  attacked  these  problems,  but  achieved 
broadly  speaking  nothing.  .  .  .  Weierstrass,   Dedekind,   and 
Cantor,  .  .  .   have    completely    solved    them.      Their    solu- 
tions ...  are  so  clear  as  to  leave  no  longer  the  slightest  doubt 
of  difficulty.     This  achievement  is  probably  the  greatest  of 
which  the  age  can  boast.  .  .  .  The  problem  of  the  infinitesimal 
was  solved  by  Weierstrass,  the  solution  of  the  other  two  was 
begun  by  Dedekind  and  definitely  accomplished  by  Cantor. 

RUSSELL,  BERTRAND. 
International  Monthly,  Vol.  4  (1901),  p.  89. 

1939.  It  was  not  till  Leibnitz  and  Newton,  by  the  discovery 
of  the  differential  calculus,  had  dispelled  the  ancient  darkness 
which  enveloped  the  conception  of  the  infinite,  and  had  clearly 
established  the  conception  of  the  continuous  and  continuous 
change,  that  a  full  and  productive  application  of  the  newly- 
found  mechanical  conceptions  made  any  progress. 

HELMHOLTZ,  H. 

Aim  and  Progress  oj  Physical  Science;  Popu- 
lar Lectures  [Flight]  (New  York,  1900),  p.  872. 


THE    CALCULUS   AND   ALLIED   TOPICS  333 

1940.  The  idea  of  an  infinitesimal  involves  no  contradic- 
tion ...  As  a  mathematician,   I  prefer  the  method  of  in- 
finitesimals to  that  of  limits,  as  far  easier  and  less  infested  with 
snares. — PIERCE,  C.  F. 

The  Law  of  Mind;  Monist,  Vol.  2  (1891-1892}, 
pp.  543,  545. 

1941.  The  chief  objection  against  all  abstract  reasonings  is 
derived  from  the  ideas  of  space  and  time;  ideas,  which,  in  com- 
mon life  and  to  a  careless  view,  are  very  clear  and  intelligible, 
but  when  they  pass  through  the  scrutiny  of  the  profound  sciences 
(and  they  are  the  chief  object  of  these  sciences)  afford  principles, 
which  seem  full  of  obscurity  and  contradiction.    No  priestly 
dogmas,  invented  on  purpose  to  tame  and  subdue  the  rebellious 
reason  of  mankind,  ever  shocked  common  sense  more  than 
the  doctrine  of  the  infinite  divisibility  of  extension,  with  its 
consequences;  as  they  are  pompously  displayed  by  all  geom- 
etricians and  metaphysicians,  with  a  kind  of  triumph  and 
exultation.     A  real  quantity,  infinitely  less   than   any  finite 
quantity,  containing  quantities  infinitely  less  than  itself,  and 
so  on  in  infinitum;  this  is  an  edifice  so  bold  and  prodigious,  that 
it  is  too  weighty  for  any  pretended  demonstration  to  support, 
because  it  shocks  the  clearest  and  most  natural  principles  of 
human  reason.    But  what  renders  the  matter  more  extraordi- 
nary, is,  that  these  seemingly  absurd  opinions  are  supported  by 
a  chain  of  reasoning,  the  clearest  and  most  natural;  nor  is  it 
possible  for  us  to  allow  the  premises  without  admitting  the  con- 
sequences.   Nothing  can  be  more  convincing  and  satisfactory 
than  all  the  conclusions  concerning  the  properties  of  circles  and 
triangles;  and  yet,  when  these  are  once  received,  how  can  we 
deny,  that  the  angle  of  contact  between  a  circle  and  its  tangent 
is  infinitely  less  than  any  rectilineal  angle,  that  as  you  may  in- 
crease the  diameter  of  the  circle  in  infinitum,  this  angle  of  con- 
tact becomes  still  less,  even  in  infinitum,  and  that  the  angle  of 
contact  between  other  curves  and  their  tangents  may  be  in- 
finitely less  than  those  between  any  circle  and  its  tangent,  and 
so  on,  in  infinitumt     The  demonstration  of  these  principles 
seems  as  unexceptionable  as  that  which  proves  the  three  angles 
of  a  triangle  to  be  equal  to  two  right  ones,  though  the  latter 


334  MEMORABILIA   MATHEMATICA 

opinion  be  natural  and  easy,  and  the  former  big  with  contradic- 
tion and  absurdity.  Reason  here  seems  to  be  thrown  into  a 
kind  of  amazement  and  suspense,  which,  without  the  suggestion 
of  any  sceptic,  gives  her  a  diffidence  of  herself,  and  of  the  ground 
on  which  she  treads.  She  sees  a  full  light,  which  illuminates 
certain  places;  but  that  light  borders  upon  the  most  profound 
darkness.  And  between  these  she  is  so  dazzled  and  confounded, 
that  she  scarcely  can  pronounce  with  certainty  and  assurance 
concerning  any  one  object. — HUME,  DAVID. 

An  Inquiry  concerning  Human  Understanding, 

Sect.  12,  part  2. 

1942.  He  who  can  digest  a  second  or  third  fluxion,  a  second  or 
third  difference,  need  not,  methinks,  be  squeamish  about  any 
point  in  Divinity. — BERKELEY,  G. 

The  Analyst,  sect.  7. 

1943.  And  what  are  these  fluxions?    The  velocities  of  evanes- 
cent increments.    And  what  are  these  same  evanescent  incre- 
ments?   They  are  neither  finite  quantities,  nor  quantities  in- 
finitely small,  nor  yet  nothing.    May  we  not  call  them  ghosts 
of  departed  quantities? — BERKELEY,  G. 

The  Analyst,  sect.  35. 

1944.  It  is  said  that  the  minutest  errors  are  not  to  be  neg- 
lected hi  mathematics;  that  the  fluxions  are  celerities,  not 
proportional  to  the  finite  increments,  though  ever  so  small; 
but  only  to  the  moments  or  nascent  increments,  whereof  the 
proportion  alone,  and  not  the  magnitude,  is  considered.    And  of 
the  aforesaid  fluxions  there  be  other  fluxions,  which  fluxions  of 
fluxions  are  called  second  fluxions.    And  the  fluxions  of  these 
second  fluxions  are  called  third  fluxions:  and  so  on,  fourth, 
fifth,  sixth,  etc.,  ad  infinitum.    Now,  as  our  Sense  is  strained  and 
puzzled  with  the  perception  of  objects  extremely  minute,  even 
so  the  Imagination,  which  faculty  derives  from  sense,  is  very 
much  strained  and  puzzled  to  frame  clear  ideas  of  the  least 
particle  of  time,  or  the  least  increment  generated  therein:  and 
much  more  to  comprehend  the  moments,  or  those  increments  of 
the  flowing  quantities  in  status  nascenti,  in  their  first  origin  or 
beginning  to  exist,  before  they  become  finite  particles.    And  it 


THE    CALCULUS   AND    ALLIED   TOPICS  335 

seems  still  more  difficult  to  conceive  the  abstracted  velocities  of 
such  nascent  imperfect  entities.  But  the  velocities  of  the 
velocities,  the  second,  third,  fourth,  and  fifth  velocities,  etc., 
exceed,  if  I  mistake  not,  all  human  understanding.  The  further 
the  mind  analyseth  and  pursueth  these  fugitive  ideas  the  more 
it  is  lost  and  bewildered;  the  objects,  at  first  fleeting  and  minute, 
soon  vanishing  out  of  sight.  Certainly,  in  any  sense,  a  second 
or  third  fluxion  seems  an  obscure  Mystery.  The  incipient 
celerity  of  an  incipient  celerity,  the  nascent  augment  of  a 
nascent  augment,  i.  e.  of  a  thing  which  hath  no  magnitude; 
take  it  in  what  light  you  please,  the  clear  conception  of  it  will,  if 
I  mistake  not,  be  found  impossible;  whether  it  be  so  or  no  I 
appeal  to  the  trial  of  every  thinking  reader.  And  if  a  second 
fluxion  be  inconceivable,  what  are  we  to  think  of  third,  fourth, 
fifth  fluxions,  and  so  on  without  end. — BERKELEY,  G. 

The  Analyst,  sect.  4- 

1945.  The  infinite  divisibility  of  finite  extension,  though  it  is 
not  expressly  laid  down  either  as  an  axiom  or  theorem  in  the 
elements  of  that  science,  yet  it  is  throughout  the  same  every- 
where supposed  and  thought  to  have  so  inseparable  and  essen- 
tial a  connection  with  the  principles  and  demonstrations  in 
Geometry,  that  mathematicians  never  admit  it  into  doubt,  or 
make  the  least  question  of  it.    And,  as  this  notion  is  the  source 
whence  do  spring  all  those  amusing  geometrical  paradoxes  which 
have  such  a  direct  repugnancy  to  the  plain  common  sense  of 
mankind,  and  are  admitted  with  so  much  reluctance  into  a 
mind  not  yet  debauched  by  learning;  so  it  is  the  principal  occa- 
sion of  all  that  nice  and  extreme  subtility  which  renders  the 
study  of  Mathematics  so  difficult  and  tedious. — BERKELEY,  G. 

On    the    Principles    of   Human    Knowledge, 
Sect.  123. 

1946.  To  avoid  misconception,  it  should  be  borne  in  mind  that 
infinitesimals  are  not  regarded  as  being  actual  quantities  in  the 
ordinary  acceptation  of  the  words,  or  as  capable  of  exact  repre- 
sentation.   They  are  introduced  for  the  purpose  of  abridgment 
and  simplification  of  our  reasonings,  and  are  an  ultimate  phase 
of  magnitude  when  it  is  conceived  by  the  mind  as  capable  of 
diminution  below  any  assigned  quantity,  however  small.  .  .  . 


336  MEMORABILIA  MATHEMATICA 

Moreover  such  quantities  are  neglected,  not,  as  Leibnitz  stated, 
because  they  are  infinitely  small  in  comparison  with  those  that 
are  retained,  which  would  produce  an  infinitely  small  error,  but 
because  they  must  be  neglected  to  obtain  a  rigorous  result;  since 
such  result  must  be  definite  and  determinate,  and  consequently 
independent  of  these  variable  indefinitely  small  quantities.  It 
may  be  added  that  the  precise  principles  of  the  infinitesimal 
calculus,  like  those  of  any  other  science,  cannot  be  thoroughly 
apprehended  except  by  those  who  have  already  studied  the 
science,  and  made  some  progress  hi  the  application  of  its  prin- 
ciples.— WILLIAMSON,  B. 

Encyclopedia  Britannica,  9th  Edition;  Article 
"Infinitesimal  Calculus,"  Sect.  12, 14- 

1947.  We  admit,  in  geometry,  not  only  infinite  magnitudes, 
that  is  to  say,  magnitudes  greater  than  any  assignable  magni- 
tude, but  infinite  magnitudes  infinitely  greater,  the  one  than  the 
other.    This  astonishes  our  dimension  of  brains,  which  is  only 
about  six  inches  long,  five  broad,  and  six  in  depth,  in  the  largest 
heads. — VOLTAIRE. 

A  Philosophical  Dictionary;  Article  "Infinity." 
(Boston,  1881). 

1948.  Infinity  is  the  land  of  mathematical  hocus  pocus. 
There  Zero  the  magician  is  king.     When  Zero  divides  any 
number  he  changes  it  without  regard  to  its  magnitude  into  the 
infinitely  small  [great?],  and  inversely,  when  divided  by  any 
number  he  begets  the  infinitely  great  [small?].    In  this  domain 
the  circumference  of  the  circle  becomes  a  straight  line,  and  then 
the  circle  can  be  squared.    Here  all  ranks  are  abolished,  for 
Zero  reduces  everything  to  the  same  level  one  way  or  another. 
Happy  is  the  kingdom  where  Zero  rules! — CARUS,  PAUL. 

Logical  and  Mathematical  Thought;  Monist, 
Vol.  20  (1909-1910),  p.  69. 

1949.  Great  fleas  have  little  fleas  upon  their  backs  to  bite  'em, 
And  little  fleas  have  lesser  fleas,  and  so  ad  infinitum. 
And  the  great  fleas  themselves,  hi  turn,  have  greater 

fleas  to  go  on; 

While  these  again  have  greater  still,  and  greater  still, 
and  so  on. — DE  MORGAN,  A. 

Budget  of  Paradoxes  (London,  1872),  p.  877. 


THE    CALCULUS   AND   ALLIED    TOPICS  337 

1950.  We  have  adroitly  defined  the  infinite  in  arithmetic  by  a 
loveknot,  in  this  manner  °° ;  but  we  possess  not  therefore  the 
clearer  notion  of  it. — VOLTAIRE. 

A  Philosophical  Dictionary;  Article  "Infinity" 
(Boston,  1881). 

1951.  I  protest  against  the  use  of  infinite  magnitude  as  some- 
thing completed,  which  in  mathematics  is  never  permissible. 
Infinity  is  merely  a  facon  de  parler,  the  real  meaning  being  a 
limit  which  certain  ratios  approach  indefinitely  near,  while 
others  are  permitted  to  increase  without  restriction. — GAUSS. 

Brief  an  Schumacher  (1881);  Werke,  Bd.  8 
p.  216. 

1952.  In  spite  of  the  essential  difference  between  the  concep- 
tions of  the  potential  and  the  actual  infinite,  the  former  signifying 
a  variable  finite  magnitude  increasing  beyond  all  finite  limits, 
while  the  latter  is  a  fixed,  constant  quantity  lying  beyond  all 
finite  magnitudes,  it  happens  only  too  often  that  the  one  is 
mistaken  for  the  other.  .  .  .  Owing  to  a  justifiable  aversion  to 
such  illegitimate  actual  infinities  and  the  influence  of  the  modern 
epicuric-materialistic  tendency,   a  certain  horror  infiniti  has 
grown  up  in  extended  scientific  circles,  which  finds  its  classic 
expression  and  support  in  the  letter  of  Gauss  [see  1951],  yet  it 
seems  to  me  that  the  consequent  uncritical  rejection  of  the 
legitimate  actual  infinite  is  no  lesser  violation  of  the  nature  of 
things,  which  must  be  taken  as  they  are. — CANTOR,  G. 

Zum  Problem  des  actualen  Unendlichen;  Natur 
und  Offenbarung,  Bd.  82  (1886),  p.  226. 

1953.  The  Infinite  is  often  confounded  with  the  Indefinite, 
but  the  two  conceptions  are  diametrically  opposed.    Instead  of 
being  a  quantity  with  unassigned  yet  assignable  limits,  the 
Infinite  is  not  a  quantity  at  all,  since  it  neither  admits  of  aug- 
mentation nor  diminution,  having  no  assignable  limits;  it  is  the 
operation  of  continuously  withdrawing  any  limits  that  may  have 
been  assigned :  the  endless  addition  of  new  quantities  to  the  old : 
the  flux  of  continuity.    The  Infinite  is  no  more  a  quantity  than 
Zero  is  a  quantity.    If  Zero  is  the  sign  of  a  vanished  quantity,  the 


338  MEMORABILIA   MATHEMATICA 

Infinite  is  a  sign  of  that  continuity  of  Existence  which  has  been 
ideally  divided  into  discrete  parts  in  the  affixing  of  limits. 

LEWES,  G.  H. 

Problems  of  Life  and  Mind  (Boston,  1875), 
Vol.  2,  p.  884. 

1954.  A  great  deal  of  misunderstanding  is  avoided  if  it  be 
remembered  that  the  terms  infinity,  infinite,  zero,  infinitesimal 
must  be  interpreted  in  connexion  with  their  context,  and  ad- 
mit a  variety  of  meanings  according  to  the  way  in  which  they 
are  defined. — MATHEWS,  G.  B. 

Theory  of  Numbers  (Cambridge,  1892),  Part  1, 
sect.  104. 

1955.  This  further  is  observable  in  number,  that  it  is  that 
which  the  mind  makes  use  of  in  measuring  all  things  that  by  us 
are  measurable,  which  principally  are  expansion  and  duration; 
and  our  idea  of  infinity,  even  when  applied  to  those,  seems  to  be 
nothing  but  the  infinity  of  number.     For  what  else  are  our 
ideas  of  Eternity  and  Immensity,  but  the  repeated  additions  of 
certain  ideas  of  imagined  parts  of  duration  and  expansion,  with 
the  infinity  of  number;  in  which  we  can  come  to  no  end  of 
addition? — LOCKE,  JOHN. 

An  Essay  concerning  Human  Understanding, 
Bk.  2,  chap.  16,  sect.  8. 

1956.  But  of  all  other  ideas,  it  is  number,  which  I  think 
furnishes  us  with  the  clearest  and  most  distinct  idea  of  infinity 
we  are  capable  of. — LOCKE,  JOHN. 

An  Essay  concerning  Human  Understanding, 
Bk.  2,  chap.  17,  sect.  9. 

1957.  Willst  du  ins  Unendliche  schreiten? 
Geh  nur  im  Endlichen  nach  alien  Seiten! 
Willst  du  dich  am  Ganzen  erquicken, 

So  musst  du  das  Ganze  im  Kleinsten  erblicken. 

GOETHE. 
Gott,  Gemut  und  Welt  (1815). 

[Would'st  thou  the  infinite  essay? 
The  finite  but  traverse  in  every  way. 
Would'st  in  the  whole  delight  thy  heart? 
Learn  to  discern  the  whole  in  its  minutest  part.] 


THE    CALCULUS   AND   ALLIED    TOPICS  339 

1958.  Ich  haufe  ungeheure  Zahlen, 
Gebiirge  Millionen  auf , 

Ich  setze  Zeit  auf  Zeit  und  Welt  auf  Welt  zu  Hauf, 

Und  wenn  ich  von  der  grausen  Hoh ' 

Mit  Schwindeln  wieder  nach  dir  seh,' 

1st  alle  Macht  der  Zahl,  vermehrt  zu  tausendmalen, 

Noch  nicht  ein  Theil  von  dir. 

Ich  zieh '  sie  ab,  und  du  liegst  ganz  vor  mir. 

HALLER,  ALBR.  VON. 
Quoted    in   Hegel:    Wissenschaft   der    Logik, 
Buch  1,  Abschnitt  2,  Kap.  2,  C,  b. 

[Numbers  upon  numbers  pile, 

Mountains  millions  high, 

Time  on  time  and  world  on  world  amass, 

Then,  if  from  the  dreadful  hight,  alas ! 

Dizzy-brained,  I  turn  thee  to  behold, 

All  the  power  of  number,  increased  thousandfold, 

Not  yet  may  match  thy  part. 

Subtract  what  I  will,  wholly  whole  thou  art.] 

1959.  A  collection  of  terms  is  infinite  when  it  contains  as 
parts  other  collections  which  have  just  as  many  terms  in  it  as  it 
has.    If  you  can  take  away  some  of  the  terms  of  a  collection, 
without  diminishing  the  number  of  terms,  then  there  is  an  in- 
finite number  of  terms  in  the  collection  — RUSSELL,  BERTRAND. 

International  Monthly,  Vol.  4  (1901),  p.  98. 

1960.  An   assemblage    (ensemble,    collection,   group,   mani- 
fold) of  elements  (things,  no  matter  what)  is  infinite  or  finite 
according  as  it  has  or  has  not  a  part  to  which  the  whole  is  just 
equivalent  in  the  sense  that  between  the  elements  composing  that 
part  and  those  composing  the  whole  there  subsists  a  unique  and 
reciprocal  (one-to-one)  correspondence. — KEYSER,  C.  J. 

The   Axioms   of  Infinity;   Hibbert  Journal, 
Vol.  2  (1903-1904),  P-  589. 

1961.  Whereas  in  former  times  the  Infinite  betrayed  its  pres- 
ence not  indeed  to  the  faculties  of  Logic  but  only  to  the  spiritual 
Imagination  and  Sensibility,  mathematics  has  shown  .  .  .  that 


340  MEMORABILIA   MATHEMATICA 

the  structure  of  Transfinite  Being  is  open  to  exploration  by  the 
organon  of  Thought. — KEYSER,  C.  J. 

Lectures  on  Science,  Philosophy  and  Art  (New 
York,  1908),  p.  42. 

1962.  The  mathematical  theory  of  probability  is  a  science 
which  aims  at  reducing  to  calculation,  where  possible,  the 
amount  of  credence  due  to  propositions  or  statements,  or  to  the 
occurrence  of  events,  future  or  past,  more  especially  as  contingent 
or  dependent  upon  other  propositions  or  events  the  probability 
of  which  is  known. — CROFTON,  M.  W. 

Encyclopedia  Britannica,  9th  Edition;  Article, 
"Probability." 

1963.  The  theory  of  probabilities  is  at  bottom  nothing  but 
common  sense  reduced  to  calculus;  it  enables  us  to  appreciate 
with  exactness  that  which  accurate  minds  feel  with  a  sort  of 
instinct  for  which  ofttimes  they  are  unable  to  account.    If  we 
consider  the  analytical  methods  to  which  this  theory  has  given 
birth,  the  truth  of  the  principles  on  which  it  is  based,  the  fine 
and  delicate  logic  which  their  employment  in  the  solution  of 
problems  requires,  the  public  utilities  whose  establishment  rests 
upon  it,  the  extension  which  it  has  received  and  which  it  may 
still  receive  through  its  application  to  the  most  important 
problems  of  natural  philosophy  and  the  moral  sciences;  if 
again  we  observe  that,  even  in  matters  which  cannot  be  sub- 
mitted to  the  calculus,  it  gives  us  the  surest  suggestions  for  the 
guidance  of  our  judgments,  and  that  it  teaches  us  to  avoid  the 
illusions  which  often  mislead  us,  then  we  shall  see  that  there  is  no 
science  more  worthy  of  our  contemplations  nor  a  more  useful 
one  for  admission  to  our  system  of  public  education. — LAPLACE. 

Theorie  Analytique  des  Probabilities,  Introduc- 
tion; Oeuvres,  t.  7  (Paris,  1886),  p.  158. 

1964.  It  is  a  truth  very  certain  that,  when  it  is  not  in  our 
power  to  determine  what  is  true,  we  ought  to  follow  what  is 
most  probable. — DESCARTES. 

Discourse  on  Method,  Part  3. 

1965.  As  demonstration  is  the  showing  the  agreement  or 
disagreement  of  two  ideas,  by  the  intervention  of  one  or  more 


THE  CALCULUS  AND  ALLIED  TOPICS  341 

proofs,  which  have  a  constant,  immutable,  and  visible  con- 
nexion one  with  another;  so  probability  is  nothing  but  the 
appearance  of  such  an  agreement  or  disagreement,  by  the  inter- 
vention of  proofs,  whose  connexion  is  not  constant  and  immut- 
able, or  at  least  is  not  perceived  to  be  so,  and  it  is  enough  to 
induce  the  mind  to  judge  the  proposition  to  be  true  or  false, 
rather  than  contrary. — LOCKE,  JOHN. 

An  Essay  concerning  Human  Understanding, 

Bk.  4,  chap.  15,  sect.  1. 

1966.  The    difference    between    necessary    and    contingent 
truths  is  indeed  the  same  as  that  between  commensurable  and 
incommensurable  numbers.    For  the  reduction  of  commensur- 
able numbers  to  a  common  measure  is  analogous  to  the  demon- 
stration of  necessary  truths,  or  their  reduction  to  such  as  are 
identical.    But  as,  in  the  case  of  surd  ratios,  the  reduction  in- 
volves  an  infinite  process,   and  yet   approaches   a   common 
measure,  so  that  a  definite  but  unending  series  is  obtained,  so 
also  contingent  truths  require  an  infinite  analysis,  which  God 
alone  can  accomplish. — LEIBNITZ. 

Philosophische    Schriften    [Gerhardt]    Bd.    7 
(Berlin,  1890},  p.  200. 

1967.  The  theory  in  question  [theory  of  probability]  affords 
an  excellent  illustration  of  the  application  of  the  theory  of 
permutation  and  combinations  which  is  the  fundamental  part  of 
the  algebra  of  discrete  quantity;  it  forms  in  the  elementary 
parts  an  excellent  logical  exercise  in  the  accurate  use  of  terms 
and  in  the  nice  discrimination  of  shades  of  meaning;  and,  above 
all,  it  enters  into  the  regulation  of  some  of  the  most  important 
practical  concerns  of  modern  life. — CHRYSTAL,  GEORGE. 

Algebra,  Vol.  2  (Edinburgh,  1889),  chap.  36, 
sect.  1. 

1968.  There  is  possibly  no  branch  of  mathematics  at  once  so 
interesting,  so  bewildering,  and  of  so  great  practical  importance 
as  the  theory  of  probabilities.     Its  history  reveals  both  the 
wonders  that  can  be  accomplished  and  the  bounds  that  cannot 
be  transcended  by  mathematical  science.    It  is  the  link  between 
rigid  deduction  and  the  vast  field  of  inductive  science.    A  com- 
plete theory  of  probabilities  would  be  the  complete  theory  of 


342  MEMORABILIA   MATHEMATICA 

the  formation  of  belief.  It  is  certainly  a  pity  then,  that,  to 
quote  M.  Bertrand,  "one  cannot  well  understand  the  calculus  of 
probabilities  without  having  read  Laplace's  work,"  and  that 
"one  cannot  read  Laplace's  work  without  having  prepared  one- 
self for  it  by  the  most  profound  mathematical  studies." 

DAVIS,  E.  W. 

Bulletin    American    Mathematical    Society, 

Vol.  1  (1894-1895),  p.  16. 

1969.  The  most  important  questions  of  life  are,  for  the  most 
part,  really  only  problems  of  probability.    Strictly  speaking  one 
may  even  say  that  nearly  all  our  knowledge  is  problematical;  and 
in  the  small  number  of  things  which  we  are  able  to  know  with 
certainty,    even    in    the    mathematical    sciences    themselves, 
induction  and  analogy,  the  principal  means  for  discovering 
truth,  are  based  on  probabilities,  so  that  the  entire  system  of 
human  knowledge  is  connected  with  this  theory. — LAPLACE. 

Theorie  Analytique  des  Probabilities,  Introduc- 
tion; Oeuvres,  t.  7  (Paris,  1886),  p.  5. 

1970.  There  is  no  more  remarkable  feature  in  the  mathe- 
matical theory  of  probability  than  the  mariner  hi  which  it  has 
been  found  to  harmonize  with,  and  justify,  the  conclusions  to 
which  mankind  have  been  led,  not  by  reasoning,  but  by  in- 
stinct and  experience,  both  of  the  individual  and  of  the  race. 
At  the  same  time  it  has  corrected,  extended,  and  invested  them 
with  a  definiteness  and  precision  of  which  these  crude,  though 
sound,  appreciations  of  common  sense  were  till  then  devoid. 

CROFTON,  M.  W. 

Encyclopedia  Britannica,  9th  Edition;  Article 
"Probability." 

1971.  It  is  remarkable  that  a  science  [probabilities]  which 
began  with  the  consideration  of  games  of  chance,  should  have 
become  the  most  important  object  of  human  knowledge. 

LAPLACE. 

Theorie  Analytique  des  Probabilites,  Introduc- 
tion; Oeuvres,  t.  7  (Paris,  1886),  p.  152. 

1972.  Not  much  has  been  added  to  the  subject  [of  probability] 
since  the  close  of  Laplace's  career.     The  history  of  science 


THE  CALCULUS  AND  ALLIED  TOPICS  343 

records  more  than  one  parallel  to  this  abatement  of  activity. 
When  such  a  genius  has  departed,  the  field  of  his  labours  seems 
exhausted  for  the  time,  and  little  left  to  be  gleaned  by  his 
successors.  It  is  to  be  regretted  that  so  little  remains  to  us  of 
the  inner  workings  of  such  gifted  minds,  and  of  the  clue  by 
which  each  of  their  discoveries  was  reached.  The  didactic  and 
synthetic  form  in  which  these  are  presented  to  the  world  retains 
but  faint  traces  of  the  skilful  inductions,  the  keen  and  delicate 
perception  of  fitness  and  analogy,  and  the  power  of  imagina- 
tion .  .  .  which  have  doubtless  guided  such  a  master  as  Laplace 
or  Newton  in  shaping  out  such  great  designs — only  the  minor 
details  of  which  have  remained  over,  to  be  supplied  by  the 
less  cunning  hand  of  commentator  and  disciple. 

CROFTON,  M.  W. 

Encyclopedia  Britannica,  9th  Edition;  Article 

"Probability." 

1973.  The  theory  of  errors  may  be  defined  as  that  branch  of 
mathematics  which  is  concerned,  first,  with  the  expression  of  the 
resultant  effect  of  one  or  more  sources  of  error  to  which  com- 
puted and  observed  quantities  are  subject;  and,  secondly,  with 
the  determination  of  the  relation  between  the  magnitude  of  an 
error  and  the  probability  of  its  occurrence. — WOODWARD,  R.  S. 

Probability  and  Theory  of  Errors  (New  York, 
1906} ,  p.  30. 

1974.  Of  all  the  applications  of  the  doctrine  of  probability 
none  is  of  greater  utility  than  the  theory  of  errors.    In  astronomy, 
geodesy,  physics,  and  chemistry,  as  in  every  science  which  at- 
tains precision  in  measuring,  weighing,  and  computing,  a  knowl- 
edge of  the  theory  of  errors  is  indispensable.    By  the  aid  of  this 
theory  the  exact  sciences  have  made  great  progress  during  the 
nineteenth  century,  not  only  in  the  actual  determinations  of  the 
constants  of  nature,  but  also  in  the  fixation  of  clear  ideas  as  to 
the  possibilities  of  future  conquests  in  the  same  direction. 
Nothing,  for  example,  is  more  satisfactory  and  instructive  in  the 
history  of  science  than  the  success  with  which  the  unique 
method  of  least  squares  has  been  applied  to  the  problems  pre- 
sented by  the  earth  and  the  other  members  of  the  solar  system. 
So  great,  in  fact,  are  the  practical  value  and  theoretical  im- 


344  MEMORABILIA   MATHEMATICA 

portance  of  least  squares,  that  it  is  frequently  mistaken  for  the 
whole  theory  of  errors,  and  is  sometimes  regarded  as  embodying 
the  major  part  of  the  doctrine  of  probability  itself. 

WOODWARD,  R.  S. 

Probability  and  Theory  of  Errors  (New  York, 
1906),  pp.  9-10. 

1975.  Direct  and  inverse  ratios  have  been  applied  by  an 
ingenious  author  to  measure  human  affections,  and  the  moral 
worth  of  actions.  An  eminent  Mathematician  attempted  to 
ascertain  by  calculation,  the  ratio  in  which  the  evidence  of  facts 
must  decrease  in  the  course  of  time,  and  fixed  the  period  when 
the  evidence  of  the  facts  on  which  Christianity  is  founded  shall 
become  evanescent,  and  when  in  consequence  no  faith  shall  be 
found  on  the  earth. — REID,  THOMAS. 

Essays  on  the  Powers  of  the  Human  Mind 
(Edinburgh,  1812),  Vol.  2,  p.  408. 


CHAPTER  XX 

THE  FUNDAMENTAL  CONCEPTS,  TIME  AND  SPACE 

2001.  Kant's  Doctrine  of  Time. 

I.  Time   is  not   an   empirical   concept   deduced   from   any 
experience,  for  neither  co-existence  nor  succession  would  enter 
into  our  perception,  if  the  representation  of  time  were  not  given 
a  priori.    Only  when  this  representation  a  priori  is  given,  can 
we  imagine  that  certain  things  happen  at  the  same  time  (simul- 
taneously) or  at  different  times  (successively) . 

II.  Time  is  a  necessary  representation  on  which  all  intuitions 
depend.     We   cannot  take   away  tune  from  phenomena   in 
general,  though  we  can  well  take  away  phenomena  out  of  time. 
In  time  alone  is  reality  of  phenomena  possible.    All  phenomena 
may  vanish,  but  time  itself  (as  the  general  condition  of  their 
possibility)  cannot  be  done  away  with. 

III.  On  this  a  priori  necessity  depends  also  the  possibility  of 
apodictic  principles  of  the  relations  of  time,  or  of  axioms  of 
time  in  general.    Time  has  one  dimension  only;  different  times 
are  not  simultaneous,  but  successive,  while  different  spaces  are 
never  successive,  but  simultaneous.    Such  principles  cannot  be 
derived  from  experience,  because  experience  could  not  impart  to 
them  absolute  universality  nor  apodictic  certainty.  .  .  . 

IV.  Time  is  not  a  discursive,  or  what  is  called  a  general  con- 
cept, but  a  pure  form  of  sensuous  intuition.    Different  times  are 
parts  only  of  one  and  the  same  time.  .  .  . 

V.  To  say  that  time  is  infinite  means  no  more  than  that  every 
definite  quantity  of  time  is  possible  only  by  limitations  of  one 
time  which  forms  the  foundation  of  all  times.     The  original 
representation  of  time  must  therefore  be  given  as  unlimited. 
But  when  the  parts  themselves  and  every  quantity  of  an  object 
can  be  represented  as  determined  by  limitation  only,  the  whole 
representation  cannot  be  given  by  concepts  (for  in  that  case  the 
partial  representation  comes  first),  but  must  be  founded  on 

immediate  intuition. — KANT,  I. 

Critique  of  Pure  Reason  [Max  Mutter]  (New 
York,  1900),  pp.  24-25. 
345 


346  MEMORABILIA    MATHEMATICA 

2002.  Kant's  Doctrine  of  Space. 

I.  Space  is  not  an  empirical  concept  which  has  been  derived 
from  external  experience.  For  in  order  that  certain  sensations 
should  be  referred  to  something  outside  myself,  i.  e.  to  some- 
thing in  a  different  part  of  space  from  that  where  I  am;  again,  in 
order  that  I  may  be  able  to  represent  them  as  side  by  side,  that 
is,  not  only  as  different,  but  as  in  different  places,  the  representa- 
tion of  space  must  already  be  there.  .  .  . 

II.  Space  is  a  necessary  representation  a  priori,  forming  the 
very  foundation  of  all  external  intuitions.    It  is  impossible  to 
imagine  that  there  should  be  no  space,  though  one  might  very 
well  imagine  that  there  should  be  space  without  objects  to  fill  it. 
Space  is  therefore  regarded  as  a  condition  of  the  possibility  of 
phenomena,  not  as  a  determination  produced  by  them;  it  is  a 
representation  a  priori  which  necessarily  precedes  all  external 
phenomena. 

III.  On  this  necessity  of  an  a  priori  representation  of  space 
rests  the  apodictic  certainty  of  all  geometrical  principles,  and 
the  possibility  of  their  construction  a  priori.    For  if  the  intuition 
of  space  were  a  concept  gained  a  posteriori,  borrowed  from 
general  external  experience,  the  first  principles  of  mathematical 
definition  would  be  nothing  but  perceptions.    They  would  be 
exposed  to  all  the  accidents  of  perception,  and  there  being  but 
one  straight  line  between  two  points  would  not  be  a  necessity, 
but  only  something  taught  in  each  case  by  experience.    What- 
ever is  derived  from  experience  possesses  a  relative  generality 
only,  based  on  induction.    We  should  therefore  not  be  able  to 
say  more  than  that,  so  far  as  hitherto  observed,  no  space  has 
yet  been  found  having  more  than  three  dimensions. 

IV.  Space  is  not  a  discursive  or  so-called  general  concept  of 
the  relations  of  things  in  general,  but  a  pure  intuition.     For, 
first  of  all,  we  can  imagine  one  space  only,  and  if  we  speak  of 
many  spaces,  we  mean  parts  only  of  one  and  the  same  space. 
Nor  can  these  parts  be  considered  as  antecedent  to  the  one  and 
all-embracing  space  and,  as  it  were,  its  component  parts  out  of 
which  an  aggregate  is  formed,  but  they  can  be  thought  of  as 
existing  within  it  only.    Space  is  essentially  one;  its  multiplicity, 
and  therefore  the  general  concept  of  spaces  in  general,  arises  en- 
tirely from  limitations.    Hence  it  follows  that,  with  respect  to 


THE  FUNDAMENTAL  CONCEPTS  OF  TIME  AND  SPACE   347 


space,  an  intuition  a  priori,  which  is  not  empirical,  must  form 
the  foundation  of  all  conceptions  of  space.  .  .  . 

V.  Space  is  represented  as  an  infinite  given  quantity.  Now  it 
is  quite  true  that  every  concept  is  to  be  thought  as  a  representa- 
tion, which  is  contained  in  an  infinite  number  of  different 
possible  representations  (as  their  common  characteristic),  and 
therefore  comprehends  them:  but  no  concept,  as  such,  can  be 
thought  as  if  it  contained  in  itself  an  infinite  number  of  repre- 
sentations. Nevertheless,  space  is  so  thought  (for  all  parts  of 
infinite  space  exist  simultaneously).  Consequently,  the  original 
representation  of  space  is  an  intuition  a  priori,  and  not  a  con- 
cept.— KANT,  I. 

Critique  of  Pure  Reason  [Max  Muller]  (New 
York,  1900),  pp.  18-20  and  Supplement  8. 

2003.  Schopenhauer's  Predicabilia  a  priori* 


OF   TIME 

1.  There  is  but  one  time,  all 

different  times  are  parts 
of  it. 

2.  Different    tunes    are    not 

simultaneous  but  succes- 
sive. 

3.  Everything  in  tune  may  be 

thought  of  as  non-exist- 
ent, but  not  time. 

4.  Time  has  three  divisions: 

past,  present  and  future, 
which  form  two  direc- 
tions with  a  point  of  in- 
difference. 

5.  Time  is  infinitely  divisible. 

6.  Time  is  homogeneous  and  a 

continuum:  i.  e.  no  part 
is  different  from  another, 
nor  separated  by  some- 
thing which  is  not  time. 

*  Schopenhauer's  table  contains  a 
which  has  here  been  omitted. 


OF   SPACE 

1.  There  is  but  one  space,  all 

different  spaces  are  parts 
of  it. 

2.  Different    spaces    are    not 

successive  but  simultane- 
ous. 

3.  Everything  in  space  may  be 

thought  of  as  non-exist- 
ent, but  not  space. 

4.  Space  has  three  dimensions : 

height,      breadth,      and 
length. 


5.  Space  is  infinitely  divisible. 

6.  Space  is  homogeneous  and  a 

continuum:  i.  e.  no  part 
is  different  from  another, 
nor  separated  by  some- 
thing which  is  not  space, 
third  column  headed  "of  matter" 


348 


MEMORABILIA   MATHEMAUCA 


7.  Time  has  no  beginning  nor 

end,    but    all    beginning 
and  end  is  in  time. 

8.  Time  makes  counting  possi- 

ble. 

9.  Rhythm  exists  only  in  tune. 

10.  The  laws  of  time  are  a 

priori  conceptions. 

11.  Time     is     perceptible     a 

priori,  but  only  by 
means  of  a  line-image. 

12.  Time  has  no  permanence 

but  passes  the  moment 
it  is  present. 

13.  Time  never  rests. 

14.  Everything    in    tune    has 

duration. 

15.  Time  has  no  duration,  but 

all  duration  is  in  tune; 
time  is  the  persistence 
of  what  is  permanent  in 
contrast  with  its  restless 
course. 

16.  Motion  is  only  possible  in 

time. 

17.  Velocity,  the  space  being 

the  same,  is  in  the  in- 
verse ratio  of  the  time. 

18.  Time  is  not  directly  meas- 

urable by  means  of  it- 
self but  only  by  means 
of  motion  which  takes 
place  hi  both  space  and 
tune.  .  .  . 

19.  Time  is  omnipresent:  each 

part  of  it  is  everywhere. 

20.  In  time  alone  all  things  are 

successive. 


7.  Space  has  no  limits  [Gran- 

zen],  but  all  limits  are  in 
space. 

8.  Space  makes  measurement 

possible. 

9.  Symmetry   exists   only   in 

space. 

10.  The  laws  of  space  are  a 

priori  conceptions. 

11.  Space  is  immediately  per- 

ceptible a  priori. 

12.  Space  never  passes  but  is 

permanent  throughout 
all  time. 

13.  Space  never  moves. 

14.  Everything  in  space  has 

position. 

15.  Space  has  no  motion,  but 

all  motion  is  in  space; 
space  is  the  change  hi 
position  of  that  which 
moves  in  contrast  to  its 
imperturbable  rest. 

16.  Motion  is  only  possible  in 

space. 

17.  Velocity,  the  tune  being 

the  same,  is  in  the  direct 
ratio  of  the  space. 

18.  Space   is   measurable   di- 

rectly through  itself  and 
indirectly  through  mo~ 
tion  which  takes  place 
in  both  tune  and 
space.  .  .  . 

19.  Space    is     eternal:    each 

part  of  it  exists  always. 

20.  In  space  alone  all  things 

are  simultaneous. 


THE    FUNDAMENTAL    CONCEPTS   OF   TIME   AND   SPACE      349 


21.  Time  makes  possible  the 

change  of  accidents. 

22.  Each   part   of  time   con- 

tains all  substance. 

23.  Time    is    the    principium 

individuationis. 

24.  The  now  is  without  dura- 

tion. 

25.  Time  of  itself  is  empty  and 

indeterminate. 

26.  Each    moment    is    condi- 

tioned by  the  one  which 
precedes  it,  and  only  so 
far  as  this  one  has 
ceased  to  exist.  (Prin- 
ciple of  sufficient  reason 
of  being  in  time.) 

27.  Time    makes    Arithmetic 

possible. 

28.  The    simple    element    of 

Arithmetic  is  unity. 


21.  Space  makes  possible  the 

endurance  of  substance. 

22.  No  part  of  space  contains 

the  same  substance  as 
another. 

23.  Space   is   the   principium 

individuationis. 

24.  The  point  is  without  ex- 

tension. 

25.  Space  is  of  itself  empty 

and  indeterminate. 

26.  The     relation     of     each 

boundary  in  space  to 
every  other  is  deter- 
mined by  its  relation  to 
any  one.  (Principle  of 
sufficient  reason  of  be- 
ing in  space.) 

27.  Space    makes    Geometry 

possible. 

28.  The  element  of  Geometry 

is  the  point. 

SCHOPENHAUER,  A. 

Die  Welt  als  Vorstellung  und  Wille;  Werke 
(Frauenstadt)  (Leipzig,  1877),  Bd.  2,  p.  55. 


2004.  The  clear  possession  of  the  Idea  of  Space  is  the  first 
requisite  for  all  geometrical  reasoning;  and  this  clearness  of 
idea  may  be  tested  by  examining  whether  the  axioms  offer 
themselves  to  the  mind  as  evident. — WHEWELL,  WILLIAM. 

The  Philosophy  of  the  Inductive  Sciences,  Part 
1,  Bk.  2,  chap.  4,  sect.  4   (London,   1858). 


2005.  Geometrical  axioms  are  neither  synthetic  a  priori  con- 
clusions nor  experimental  facts.  They  are  conventions:  our 
choice,  amongst  all  possible  conventions,  is  guided  by  experi- 
mental facts;  but  it  remains  free,  and  is  only  limited  by  the 
necessity  of  avoiding  all  contradiction.  ...  In  other  words, 
axioms  of  geometry  are  only  definitions  in  disguise. 


350  MEMORABILIA   MATHEMATICA 

That  being  so  what  ought  one  to  think  of  this  question:  Is  the 
Euclidean  Geometry  true? 

The  question  is  nonsense.  One  might  as  well  ask  whether  the 
metric  system  is  true  and  the  old  measures  false;  whether 
Cartesian  co-ordinates  are  true  and  polar  co-ordinates  false. 

POINCARE,  H. 

Non-Euclidean   Geometry;    Nature,    Vol.    46 
(1891-1892),  p.  407. 

2006.  I  do  in  no  wise  share  this  view  [that  the  axioms  are 
arbitrary  propositions  which  we  assume  wholly  at  will,  and  that 
in  like  manner  the  fundamental  conceptions  are  in  the  end  only 
arbitrary  symbols  with  which  we  operate]  but  consider  it  the 
death  of  all  science:  in  my  judgment  the  axioms  of  geometry  are 
not   arbitrary,   but    reasonable  propositions    which  generally 
have  the  origin  in  space  intuition  and  whose  separate  content 
and  sequence  is  controlled  by  reasons  of  expediency. — KLEIN,  F. 

Elementarmathematik     vom     hoheren    Stand- 
punkte  aus  (Leipzig,  1909),  Bd.  2,  p.  884. 

2007.  Euclid's  Postulate  5  [The  Parallel  Axiom]. 

That,  if  a  straight  line  falling  on  two  straight  lines  make  the 
interior  angles  on  the  same  side  less  than  two  right  angles,  the 
two  straight  lines,  if  produced  indefinitely,  meet  on  that  side  on 

which  are  the  angles  less  than  the  two  right  angles. — EUCLID. 
The    Thirteen    Books    of   Euclid's   Elements 
[T.  L.  Heath]  Vol.  1  (Cambridge,  1908),  p.  202. 

2008.  It  must  be  admitted  that  Euclid's  [Parallel]  Axiom  is 
unsatisfactory  as  the  basis  of  a  theory  of  parallel  straight 
lines.    It  cannot  be  regarded  as  either  simple  or  self-evident, 
and  it  therefore  falls  short  of  the  essential  characteristics  of  an 
axiom.  .  .  .  — HALL,  H.  S.  and  STEVENS,  F.  H. 

Euclid's  Elements  (London,  1892),  p.  55. 

2009.  We  may  still  well  declare  the  parallel  axiom  the  sim- 
plest assumption  which  permits  us  to  represent  spatial  relations, 
and  so  it  will  be  true  generally,  that  concepts  and  axioms  are 
not  immediate  facts  of  intuition,  but  rather  the  idealizations  of 

these  facts  chosen  for  reasons  of  expediency. — KLEIN,  F. 

Elementarmathematik     vom     hoheren     Stanf- 
punkte  aus  (Leipzig,  1909),  Bd.  2,  p.  882. 


THE  FUNDAMENTAL  CONCEPTS  OF  TIME  AND  SPACE   351 

2010.  The  characteristic  features  of  our  space  are  not  necessi- 
ties of  thought,  and  the  truth  of  Euclid's  axioms,  in  so  far  as 
they  specially  differentiate  our  space  from  other  conceivable 
spaces,  must  be  established  by  experience  and  by  experience 
only. — BALL,  R.  S. 

Encyclopedia  Britannica,  9th  Edition;  Article 
"Measurement." 

2011.  Mathematical  and  physiological  researches  have  shown 
that  the  space  of  experience  is  simply  an  actual  case  of  many 
conceivable  cases,  about  whose  peculiar  properties  experience 
alone  can  instruct  us. — MACH,  ERNST. 

Popular  Scientific  Lectures   (Chicago,  1910), 
p.  205. 

2012.  The  familiar  definition:  An  axiom  is  a  self-evident 
truth,  means  if  it  means  anything,  that  the  proposition  which  we 
call  an  axiom  has  been  approved  by  us  in  the  light  of  our  ex- 
perience and  intuition.     In  this  sense  mathematics  has  no 
axioms,  for  mathematics  is  a  formal  subject  over  which  formal 
and  not  material  implication  reigns. — WILSON,  E.  B. 

Bulletin    American    Mathematical    Society, 
Vol.  2  (1904-1905),  p.  81. 

2013.  The  proof  of  self-evident  propositions  may  seem,  to  the 
uninitiated,   a  somewhat  frivolous  occupation.     To  this  we 
might  reply  that  it  is  often  by  no  means  self-evident  that  one 
obvious  proposition  follows  from  another  obvious  proposition; 
so  that  we  are  really  discovering  new  truths  when  we  prove  what 
is  evident  by  a  method  which  is  not  evident.     But  a  more 
interesting  retort  is,  that  since  people  have  tried  to  prove 
obvious  propositions,  they  have  found  that  many  of  them  are 
false.    Self-evidence  is  often  a  mere  will-o'-the-wisp,  which  is 
sure  to  lead  us  astray  if  we  take  it  as  our  guide. 

RUSSELL,  BERTRAND. 
Recent  Work  on  the  Principles  of  Mathematics; 
International  Monthly,  Vol.  4  (1901),  p.  86. 

2014.  The  problem  [of  Euclid's  Parallel  Axiom]  is  now  at  a  par 
with  the  squaring  of  the  circle  and  the  trisection  of  an  angle  by 
means  of  ruler  and  compass.    So  far  as  the  mathematical  public 


352  MEMORABILIA   MATHEMATICA 

is  concerned,  the  famous  problem  of  the  parallel  is  settled  for  all 

time. — YOUNG,  JOHN  WESLEY. 

Fundamental  Concepts  of  Algebra  and  Geome- 
try (New  York,  1911),  p.  32. 

2016.  If  the  Euclidean  assumptions  are  true,  the  constitution 
of  those  parts  of  space  which  are  at  an  infinite  distance  from  us, 
"geometry  upon  the  plane  at  infinity,"  is  just  as  well  known  as 
the  geometry  of  any  portion  of  this  room.  In  this  infinite  and 
thoroughly  well-known  space  the  Universe  is  situated  during  at 
least  some  portion  of  an  infinite  and  thoroughly  well-known 
time.  So  that  here  we  have  real  knowledge  of  something  at 
least  that  concerns  the  Cosmos;  something  that  is  true  through- 
out the  Immensities  and  the  Eternities.  That  something 
Lobatchewsky  and  his  successors  have  taken  away.  The 
geometer  of  to-day  knows  nothing  about  the  nature  of  the 
actually  existing  space  at  an  infinite  distance;  he  knows  nothing 
about  the  properties  of  this  present  space  in  a  past  or  future 
eternity.  He  knows,  indeed,  that  the  laws  assumed  by  Euclid 
are  true  with  an  accuracy  that  no  direct  experiment  can  ap- 
proach, not  only  in  this  place  where  we  are,  but  in  places  at  a 
distance  from  us  that  no  astronomer  has  conceived;  but  he 
knows  this  as  of  Here  and  Now;  beyond  this  range  is  a  There  and 
Then  of  which  he  knows  nothing  at  present,  but  may  ultimately 
come  to  know  more. — CLIFFORD,  W.  K. 

Lectures  and  Essays  (New  York,  1901),  Vol.  1, 

pp.  S58-859. 

2016.  The  truth  is  that  other  systems  of  geometry  are  possi- 
ble, yet  after  all,  these  other  systems  are  not  spaces  but  other 
methods  of  space  measurements.     There  is  one  space  only, 
though  we  may  conceive  of  many  different  manifolds,  which  are 
contrivances  or  ideal  constructions  invented  for  the  purpose  of 
determining  space. — CARUS,  PAUL. 

Science,  Vol.  18  (1903),  p.  106. 

2017.  As  I  have  formerly  stated  that  from  the  philosophic 
side  Non-Euclidean  Geometry  has  as  yet  not  frequently  met 
with  full  understanding,  so  I  must  now  emphasize  that  it  is 
universally  recognized  in  the  science  of  mathematics;  indeed, 


THE    FUNDAMENTAL   CONCEPTS    OF   TIME   AND    SPACE       353 

for  many  purposes,  as  for  instance  in  the  modern  theory  of 
functions,  it  is  used  as  an  extremely  convenient  means  for  the 
visual  representation  of  highly  complicated  arithmetical  rela- 
tions.— KLEIN,  F. 

Elementarmathematik  vom  hoheren  Stand- 
punkte  aus  (Leipzig,  1909),  Bd.  2,  p.  377. 

2018.  Everything  in  physical  science,  from  the  law  of  gravita- 
tion to  the  building  of  bridges,  from  the  spectroscope  to  the  art 
of  navigation,   would  be  profoundly  modified  by  any  con- 
siderable inaccuracy  in  the  hypothesis  that  our  actual  space  is 
Euclidean.    The  observed  truth  of  physical  science,  therefore, 
constitutes  overwhelming  empirical  evidence  that  this  hypothe- 
sis is  very  approximately  correct,  even  if  not  rigidly  true. 

RUSSELL,  BERTBAND. 

Foundations  of  Geometry  (Cambridge,  1897), 
p.  6. 

2019.  The  most  suggestive  and  notable  achievement  of  the 
last  century  is  the  discovery  of  Non-Euclidean  geometry. 

HILBERT,  D. 

Quoted  by  G.  D.  Fitch  in  Manning's  "The 
Fourth  Dimension  Simply  Explained,"  (New 
York,  1910),  p.  58. 

2020.  Non-Euclidean  geometry — primate  among  the  eman- 
cipators of  the  human  intellect.  .  .  . — KEYSER,  C.  J. 

The  Foundations  of  Mathematics;  Science 
History  of  the  Universe,  Vol.  8  (New  York, 
1909),  p.  192. 

2021.  Every  high  school  teacher  [Gymnasial-lehrer]  must  of 
necessity  know  something  about  non-euclidean  geometry,  be- 
cause it  is  one  of  the  few  branches  of  mathematics  which,  by 
means  of  certain  catch-phrases,  has  become  known  in  wider 
circles,  and  concerning  which  any  teacher  is  consequently  liable 
to  be  asked  at  any  time.    In  physics  there  are  many  such  mat- 
ters— almost   every  new   discovery   is   of  this   kind — which, 
through  certain  catch-words  have  become  topics  of  common 
conversation,  and  about  which  therefore  every  teacher  must  of 
course  be  informed.    Think  of  a  teacher  of  physics  who  knows 


354  MEMORABILIA   MATHEMATICA 

nothing  of  Roentgen  rays  or  of  radium;  no  better  impression 
would  be  made  by  a  mathematician  who  is  unable  to  give  in- 
formation concerning  non-euclidean  geometry. — KLEIN,  F. 

Elementarmathematik    vom    hoheren    Stand- 
punkte  aus  (Leipzig,  1909),  Bd.  2,  p.  878. 

2022.  What  Vesalius  was  to  Galen,  what  Copernicus  was  to 
Ptolemy,  that  was  Lobatchewsky  to  Euclid.    There  is,  indeed, 
a  somewhat  instructive  parallel  between  the  last  two  cases. 
Copernicus  and  Lobatchewsky  were  both  of  Slavic  origin. 
Each  of  them  has  brought  about  a  revolution  in  scientific 
ideas  so  great  that  it  can  only  be  compared  with  that  wrought 
by  the  other.    And  the  reason  of  the  transcendent  importance  of 
these  two  changes  is  that  they  are  changes  in  the  conception  of 
the  Cosmos.  .  .  .  And  hi  virtue  of  these  two  revolutions  the 
idea  of  the  Universe,  the  Macrocosm,  the  All,  as  subject  of 
human  knowledge,  and  therefore  of  human  interest,  has  fallen 
to  pieces. — CLIFFORD,  W.  K. 

Lectures  and  Essays  (New  York,  1901),  Vol.  1, 
pp.  356,  358. 

2023.  I  am  exceedingly  sorry  that  I  have  failed  to  avail  my- 
self of  our  former  greater  proximity  to  learn  more  of  your  work 
on  the  foundations  of  geometry;  it  surely  would  have  saved  me 
much  useless  effort  and  given  me  more  peace,  than  one  of  my 
disposition  can  enjoy  so  long  as  so  much  is  left  to  consider  in  a 
matter  of  this  kind.    I  have  myself  made  much  progress  in  this 
matter  (though  my  other  heterogeneous  occupations  have  left 
me  but  little  time  for  this  purpose) ;  though  the  course  which 
I  have  pursued  does  not  lead  as  much  to  the  desired  end,  which 
you  assure  me  you  have  reached,  as  to  the  questioning  of  the 
truth  of  geometry.    It  is  true  that  I  have  found  much  which 
many  would  accept  as  proof,  but  which  in  my  estimation  proves 
nothing,  for  instance,  if  it  could  be  shown  that  a  rectilinear 
triangle  is  possible,  whose  area  is  greater  than  that  of  any  given 
surface,  then  I  could  rigorously  establish  the  whole  of  geometry. 
Now  most  people,  no  doubt,  would  grant  this  as  an  axiom, 
but  not  I;  it  is  conceivable  that,  however  distant  apart  the 
vertices  of  the  triangle  might  be  chosen,  its  area  might  yet 


THE  FUNDAMENTAL  CONCEPTS  OF  TIME  AND  SPACE   355 

always  be  below  a  certain  limit.    I  have  found  several  other 
such  theorems,  but  none  of  them  satisfies  me. — GAUSS. 

Letter  to  Bolyai  (1799);  Werke,  Bd.  8  (Gottin- 
gen,  1900),  p.  159. 

2024.  On  the  supposition  that  Euclidean  geometry  is  not 
valid,  it  is  easy  to  show  that  similar  figures  do  not  exist;  in  that 
case  the  angles  of  an  equilateral  triangle  vary  with  the  side  in 
which  I  see  no  absurdity  at  all.    The  angle  is  a  function  of  the 
side  and  the  sides  are  functions  of  the  angle,  a  function  which,  of 
course,  at  the  same  time  involves  a  constant  length.    It  seems 
somewhat  of  a  paradox  to  say  that  a  constant  length  could  be 
given  a  priori  as  it  were,  but  in  this  again  I  see  nothing  incon- 
sistent.   Indeed,  it  would  be  desirable  that  Euclidean  geometry 
were  not  valid,  for  then  we  should  possess  a  general  a  priori 
standard  of  measure. — GAUSS. 

Letter  to  Gerling  (1816);  Werke,  Bd.  8  (Gottin- 
gen,  1900),  p.  169. 

2025.  I  am  convinced  more  and  more  that  the  necessary 
truth  of  our  geometry  cannot  be  demonstrated,  at  least  not  by 
the  human  intellect  to  the  human  understanding.    Perhaps  in 
another  world  we  may  gain  other  insights  into  the  nature  of 
space  which  at  present  are  unattainable  to  us.    Until  then  we 
must  consider  geometry  as  of  equal  rank  not  with  arithmetic, 
which  is  purely  a  priori,  but  with  mechanics. — GAUSS. 

Letter  to  Olbers  (1817);  Werke,  Bd.  8  (Gottin- 
gen,  1900),  p.  177. 

2026.  There  is  no  doubt  that  it  can  be  rigorously  estab- 
lished that  the  sum  of  the  angles  of  a  rectilinear  triangle  cannot 
exceed  180°.    But  it  is  otherwise  with  the  statement  that  the 
sum  of  the  angles  cannot  be  less  than  180°;  this  is  the  real 
Gordian  knot,  the  rocks  which  cause  the  wreck  of  all.  ... 
I  have  been  occupied  with  the  problem  over  thirty  years  and  I 
doubt  if  anyone  has  given  it  more  serious  attention,  though  I 
have  never  published  anything  concerning  it.    The  assumption 
that  the  angle  sum  is  less  than  180°  leads  to  a  peculiar  geometry, 
entirely  different  from  the  Euclidean,  but  throughout  consistent 
with  itself.    I  have  developed  this  geometry  to  my  own  satisfac- 


356  MEMORABILIA   MATHEMATICA 

tion  so  that  I  can  solve  every  problem  that  arises  in  it  with  the 
exception  of  the  determination  of  a  certain  constant  which  can- 
not be  determined  a  priori.  The  larger  one  assumes  this  con- 
stant the  more  nearly  one  approaches  the  Euclidean  geometry, 
an  infinitely  large  value  makes  the  two  coincide.  The  theorems 
of  this  geometry  seem  hi  part  paradoxical,  and  to  the  un- 
practiced  absurd;  but  on  a  closer  and  calm  reflection  it  is  found 
that  in  themselves  they  contain  nothing  impossible.  .  .  .  All 
my  efforts  to  discover  some  contradiction,  some  inconsistency  in 
this  Non-Euclidean  geometry  have  been  fruitless,  the  one  thing 
in  it  that  seems  contrary  to  reason  is  that  space  would  have  to 
contain  a  definitely  determinate  (though  to  us  unknown)  linear 
magnitude.  However,  it  seems  to  me  that  notwithstanding  the 
meaningless  word-wisdom  of  the  metaphysicians  we  know  really 
too  little,  or  nothing,  concerning  the  true  nature  of  space  to 
confound  what  appears  unnatural  with  the  absolutely  impossible. 
Should  Non-Euclidean  geometry  be  true,  and  this  constant  bear 
some  relation  to  magnitudes  which  come  within  the  domain  of 
terrestrial  or  celestial  measurement,  it  could  be  determined  a 

posteriori. — GAUSS. 

Letter  to  Taurinus  (1824);  Werke,  Bd.  8  (Got- 
tingen,  1900),  p.  187. 

2027.  There  is  also  another  subject,  which  with  me  is  nearly 
forty  years  old,  to  which  I  have  again  given  some  thought 
during  leisure  hours,  I  mean  the  foundations  of  geometry.  .  .  . 
Here,  too,  I  have  consolidated  many  things,  and  my  conviction 
has,  if  possible  become  more  firm  that  geometry  cannot  be  com- 
pletely established  on  a  priori  grounds.  In  the  mean  tune  I  shall 
probably  not  for  a  long  time  yet  put  my  very  extended  investiga- 
tions concerning  this  matter  hi  shape  for  publication,  possibly 
not  while  I  live,  for  I  fear  the  cry  of  the  Boeotians  which  would 
arise  should  I  express  my  whole  view  on  this  matter. — It  is 
curious  too,  that  besides  the  known  gap  in  Euclid's  geometry, 
to  fill  which  all  efforts  till  now  have  been  in  vain,  and  which  will 
never  be  filled,  there  exists  another  defect,  which  to  my  knowl- 
edge no  one  thus  far  has  criticised  and  which  (though  possible) 
it  is  by  no  means  easy  to  remove.  This  is  the  definition  of  a 
plane  as  a  surface  which  wholly  contains  the  line  joining  any 


THE  FUNDAMENTAL  CONCEPTS  OF  TIME  AND  SPACE   357 

two  points.  This  definition  contains  more  than  is  necessary  to 
the  determination  of  the  surface,  and  tacitly  involves  a  theorem 
which  demands  proof. — GAUSS. 

Letter  to  Bessel  (1829);  Werke,  Bd.  8  (Gottin- 

gen,  1900},  p.  200. 

2028.  I  will  add  that  I  have  recently  received  from  Hungary  a 
little  paper  on  Non-Euclidean  geometry,  in  which  I  rediscover 
all  my  own  ideas  and  results  worked  out  with  great  elegance,  .  .  . 
The  writer  is  a  very  young  Austrian  officer,  the  son  of  one  of 
my  early  friends,  with  whom  I  often  discussed  the  subject  in 
1798,  although  my  ideas  were  at  that  time  far  removed  from  the 
development  and  maturity  which  they  have  received  through 
the  original  reflections  of  this  young  man.    I  consider  the  young 
geometer  v.  Bolyai  a  genius  of  the  first  rank. — GAUSS. 

Letter  to  Gerling  (1832};  Werke,  Bd.  8  (Gottin- 
gen,  1900),  p.  221. 

2029.  Think  of  the  image  of  the  world  in  a  convex  mir- 
ror. ...  A  well-made  convex  mirror  of  moderate  aperture 
represents  the  objects  in  front  of  it  as  apparently  solid  and  in 
fixed  positions  behind  its  surface.    But  the  images  of  the  distant 
horizon  and  of  the  sun  in  the  sky  lie  behind  the  mirror  at  a 
limited  distance,  equal  to  its  focal  length.    Between  these  and 
the  surface  of  the  mirror  are  found  the  images  of  all  the  other 
objects  before  it,  but  the  images  are  diminished  and  flattened  in 
proportion  to  the  distance  of  their  objects  from  the  mirror.  .  .  . 
Yet  every  straight  line  or  plane  in  the  outer  world  is  represented 
by  a  straight  [?]  line  or  plane  in  the  image.    The  image  of  a 
man  measuring  with  a  rule  a  straight  line  from  the  mirror, 
would  contract  more  and  more  the  farther  he  went,  but  with 
his  shrunken  rule  the  man  in  the  image  would  count  out  exactly 
the  same  number  of  centimeters  as  the  real  man.     And,  in 
general,  all  geometrical  measurements  of  lines  and  angles  made 
with  regularly  varying  images  of  real  instruments  would  yield 
exactly  the  same  results  as  in  the  outer  world,  all  lines  of  sight 
in  the  mirror  would  be  represented  by  straight  lines  of  sight  in 
the  mirror.    In  short,  I  do  not  see  how  men  in  the  mirror  are  to 
discover  that  their  bodies  are  not  rigid  solids  and  their  expe- 
riences good  examples  of  the  correctness  of  Euclidean  axioms. 


358  MEMORABILIA  MATHEMATICA 

But  if  they  could  look  out  upon  our  world  as  we  look  into  theirs 
without  overstepping  the  boundary,  they  must  declare  it  to  be  a 
picture  in  a  spherical  mirror,  and  would  speak  of  us  just  as  we 
speak  of  them;  and  if  two  inhabitants  of  the  different  worlds 
could  communicate  with  one  another,  neither,  as  far  as  I  can 
see,  would  be  able  to  convince  the  other  that  he  had  the  true, 
the  other  the  distorted,  relation.  Indeed  I  cannot  see  that  such 
a  question  would  have  any  meaning  at  all,  so  long  as  mechanical 
considerations  are  not  mixed  up  with  it. — HELMHOLTZ,  H. 

On  the  Origin  and  Significance  of  Geometrical 
Axioms;  Popular  Scientific  Lectures,  second 
series  (New  York,  1881),  pp.  57-59. 

2030.  That  space  conceived  of  as  a  locus  of  points  has  but 
three  dimensions  needs  no  argument  from  the  mathematical 
point  of  view;  but  just  as  little  can  we  from  this  point  of  view 
prevent  the  assertion  that  space  has  really  four  or  an  infinite 
number  of  dimensions  though  we  perceive  only  three.     The 
theory  of  multiply-extended  manifolds,  which  enters  more  and 
more  into  the  foreground  of  mathematical  research,  is  from  its 
very  nature  perfectly  independent  of  such  an  assertion.    But 
the  form  of  expression,  which  this  theory  employs,  has  indeed 
grown  out  of  this  conception.    Instead  of  referring  to  the  in- 
dividuals of  a  manifold,  we  speak  of  the  points  of  a  higher 
space,  etc.    In  itself  this  form  of  expression  has  many  advan- 
tages, in  that  it  facilitates  comprehension  by  calling  up  geometri- 
cal intuition.    But  it  has  this  disadvantage,  that  hi  extended 
circles,   investigations  concerning  manifolds  of   any  number 
of  dimensions   are   considered  singular  alongside  the  above- 
mentioned  conception   of  space.     This  view  is  without   the 
least  foundation.    The  investigations  in  question  would  indeed 
find  immediate  geometric  applications  if  the  conception  were 
valid  but  its  value  and  purpose,  being  independent  of  this  con- 
ception, rests  upon  its  essential  mathematical  content. 

KLEIN,  F. 
Mathematische  Annakn,  Bd.  43  (1893),  p.  95. 

2031.  We  are  led  naturally  to  extend  the  language  of  geom- 
etry to  the  case  of  any  number  of  variables,  still  using  the 
word  point  to  designate  any  system  of  values  of  n  variables  (the 


THE   FUNDAMENTAL   CONCEPTS   OF   TIME   AND    SPACE       359 

coordinates  of  the  point),  the  word  space  (of  n  dimensions)  to 
designate  the  totality  of  all  these  points  or  systems  of  values, 
curves  or  surface  to  designate  the  spread  composed  of  points 
whose  coordinates  are  given  functions  (with  the  proper  restric- 
tions) of  one  or  two  parameters  (the  straight  line  or  plane,  when 
they  are  linear  fractional  functions  with  the  same  denominator), 
etc.  Such  an  extension  has  come  to  be  a  necessity  in  a  large 
number  of  investigations,  in  order  as  well  to  give  them  the 
greatest  generality  as  to  preserve  in  them  the  intuitive  charac- 
ter of  geometry.  But  it  has  been  noted  that  in  such  use  of 
geometric  language  we  are  no  longer  constructing  truly  a 
geometry,  for  the  forms  that  we  have  been  considering  are 
essentially  analytic,  and  that,  for  example,  the  general  pro- 
jective  geometry  constructed  in  this  way  is  in  substance  noth- 
ing more  than  the  algebra  of  linear  transformations. 

SEGRE,  CORRADI. 

Rivista  di  Matematica,  Vol.  1   (1891),  p.  59. 
[J.  W.  Young.] 

2032.  Those  who  can,  in  common  algebra,  find  a  square 
root  of  - 1,  will  be  at  no  loss  to  find  a  fourth  dimension  in 
space  in  which  ABC  may  become  ABCD:  or,  if  they  cannot 
find  it,  they  have  but  to  imagine  it,  and  call  it  an  impossible 
dimension,  subject  to  all  the  laws  of  the  three  we  find  possible. 
And  just  as  y-1  in  common  algebra,  gives  all  its  significant 
combinations  true,  so  would  it  be  with  any  number  of  dimensions 
of  space  which  the  speculator  might  choose  to  call  into  impossible 
existence  — DE  MORGAN,  A. 

Trigonometry  and  Double  Algebra  (London, 
1849),  Part  2,  chap.  8. 

2033.  The  doctrine  of  non-Euclidean  spaces  and  of  hyper- 
spaces  in  general  possesses  the  highest  intellectual  interest,  and 
it  requires  a  far-sighted  man  to  foretell  that  it  can  never  have 
any  practical  importance. — SMITH,  W.  B. 

Introductory  Modern  Geometry   (New   York, 
1893),  p.  274. 

2034.  According  to  his  frequently  expressed  view,   Gauss 
considered  the  three  dimensions  of  space  as  specific  peculiarities 


360  MEMORABILIA   MATHEMATICA 

of  the  human  soul;  people,  which  are  unable  to  comprehend 
this,  he  designated  in  his  humorous  mood  by  the  name  Boeo- 
tians. We  could  imagine  ourselves,  he  said,  as  beings  which 
are  conscious  of  but  two  dimensions;  higher  beings  might  look  at 
us  in  a  like  manner,  and  continuing  jokingly,  he  said  that  he 
had  laid  aside  certain  problems  which,  when  in  a  higher  state 
of  being,  he  hoped  to  investigate  geometrically. 

SARTORITTS,  W.  v.  WALTERSHAUSEN. 
Gauss  zum  Gedachtniss  (Leipzig,  1856),  p.  81. 

2036.  There  is  many  a  rational  logos,  and  the  mathematician 
has  high  delight  hi  the  contemplation  of  inconsistent  systems  of 
consistent  relationships.  There  are,  for  example,  a  Euclidean 
geometry  and  more  than  one  species  of  non-Euclidean.  As 
theories  of  a  given  space,  these  are  not  compatible.  If  our 
universe  be,  as  Plato  thought,  and  nature-science  takes  for 
granted,  a  space-conditioned,  geometrised  affair,  one  of  these 
geometries  may  be,  none  of  them  may  be,  not  all  of  them  can 
be,  valid  hi  it.  But  in  the  vaster  world  of  thought,  all  of  them 
are  valid,  there  they  co-exist,  and  interlace  among  themselves 
and  others,  as  differing  component  strains  of  a  higher,  strictly 
supernatural,  hypercosmic,  harmony. — KEYSER,  C.  J. 

The  Universe  and  Beyond;  Hibbert  Journal, 
Vol.  3  (1904-1905),  p.  818. 

2036.  The  introduction  into  geometrical  work  of  conceptions 
such  as  the  infinite,  the  imaginary,  and  the  relations  of  hyper- 
space,  none  of  which  can  be  directly  imagined,  has  a  psychologi- 
cal significance  well  worthy  of  examination.    It  gives  a  deep 
insight  into  the  resources  and  working  of  the  human  mind. 
We  arrive  at  the  borderland  of  mathematics  and  psychology. 

MERZ,  J.  T. 

History  of  European  Thought  in  the  Nine- 
teenth Century  (Edinburgh  and  London,  1903), 
p.  716. 

2037.  Among  the  splendid  generalizations  effected  by  mod- 
ern mathematics,  there  is  none  more  brilliant  or  more  inspiring 
or  more  fruitful,  and  none  more  commensurate  with  the  limit- 
less immensity  of  being  itself,  than  that  which  produced  the 


THE  FUNDAMENTAL  CONCEPTS  OF  TIME  AND  SPACE   361 

great  concept  designated  .  .  .  hyperspace  or  multidimensional 
space. — KEYSER,  C.  J. 

Mathematical  Emancipations;  Monist,  Vol.  16 

(1906),  p.  65. 

2038.  The  great  generalization  [of  hyperspace]  has  made  it 
possible  to  enrich,  quicken  and  beautify  analysis  with  the  terse, 
sensuous,  artistic,  stimulating  language  of  geometry.    On  the 
other  hand,  the  hyperspaces  are  in  themselves  immeasurably 
interesting  and  inexhaustibly  rich  fields  of  research.    Not  only 
does  the  geometrician  find  light  in  them  for  the  illumination  of 
otherwise  dark  and  undiscovered  properties  of  ordinary  spaces 
of  intuition,  but  he  also  discovers  there  wondrous  structures 
quite  unknown  to  ordinary  space.  ...  It  is  by  creation  of 
hyperspaces  that  the  rational  spirit  secures  release  from  limita- 
tion.   In  them  it  lives  ever  joyously,  sustained  by  an  unfailing 
sense  of  infinite  freedom. — KEYSER,  C.  J. 

Mathematical  Emancipations;  Monist,  Vol.  16 
(1906),  p.  83. 

2039.  Mathematicians  who  busy  themselves  a  great  deal 
with  the  formal  theory  of  four-dimensional  space,  seem  to  ac- 
quire a  capacity  for  imagining  this  form  as  easily  as  the  three- 
dimensional  form  with  which  we  are  all  familiar. — OSTWALD,  W. 

Natural    Philosophy    [Seltzer],    (New    York, 
1910),  p.  77. 

2040.  Fuchs.  Was  soil  ich  nun  aber  denn  studieren? 
Meph.  Ihr  konnt  es  mit  analytischer  Geometrie  pro- 

bieren. 

Da  wird  der  Raum  euch  wohl  dressiert, 
In  Coordinaten  eingeschniirt, 
Dass  ihr  nicht  etwa  auf  gut  Gliick 
Von  der  Figur  gewinnt  ein  Stuck. 
Dann  lehret  man  euch  manchen  Tag, 
Dass,  was  ihr  sonst  auf  einen  Schlag 
Construiertet  im  Raume  frei, 
Eine  Gleichung  dazu  notig  sei. 
Zwar  war  dem  Menschen  zu  seiner  Erbauung 
Die  dreidimensionale  Raumanschauung, 


362  MEMORABILIA   MATHEMATICA 

Dass  er  sieht,  was  urn  ihn  passiert, 

Und  die  Figuren  sich  construiert — 

Der  Analytiker  tritt  herein 

Und  beweist,  das  konnte  auch  anders  sein. 

Gleichungen,  die  auf  dem  Papiere  stehn, 

Die  mttsst'  man  auch  konnen  im  Raume  sehn; 

Und  konnte  man's  nicht  construieren, 

Da  miisste  man's  anders  definieren. 

Denn  was  man   formt  nach   Zahlengesetzen 

Miisst'  uns  auch  geometrisch  erletzen. 

Drum  in  den  unendlich  fernen  beiden 

Imaginaren   Punkten  miissen  sich  schneiden 

Alle  Kreise  fein  sauberlich, 

Auch  Parallelen,  die  treffen  sich, 

Und  im  Raume  kann  man  daneben 

Allerlei  Kriimmungsmasse  erleben. 

Die  Formeln  sind  alle  wahr  und  schon, 

Warum  sollen  sie  nicht  zu  deuten  gehn? 

Da  preisen's  die  Schiller  aller  Orten, 

Dass  das  Gerade  ist  krumm  geworden. 

Nicht-Euklidisch  nennt's  die  Geometric, 

Spotted  ihrer  selbst,  und  weiss  nicht  wie. 

Fuchs.  Kann  euch  nicht  eben  ganz  verstehn. 

Meph.  Das  soil  den  Philosophen  auch  so  gehn. 
Doch  wenn  ihr  lernt  alles  reducieren 
Und  gehorig  transformieren, 
Bis  die  Formeln  den  Sinn  verlieren, 
Dann  versteht  ihr  mathematish  zu  spekulieren. 

LASSWITZ,  KURD. 

Der  Faust-Tragodie  (-n)ter  Teil;  Zeitschrift 
fur  den  math-naturw.  Unterricht,  Bd.  14 
(1888),  p.  316. 

[Fuchs.  To  what  study  then  should  I  myself  apply? 

Meph.  Begin  with  analytical  geometry. 

There  all  space  is  properly  trained, 
By  coordinates  well  restrained, 
That  no  one  by  some  lucky  assay 
Carry  some  part  of  the  figure  away. 


THE   FUNDAMENTAL   CONCEPTS   OF   TIME   AND    SPACE       363 

Next  thou'll  be  taught  to  realize, 

Constructions  won't  help  thee  to  geometrize, 

And  the  result  of  a  free  construction 

Requires  an  equation  for  proper  deduction. 

Three-dimensional  space  relation 

Exists  for  human  edification, 

That  he  may  see  what  about  him  transpires, 

And  construct  such  figures  as  he  requires. 

Enters  the  analyst.    Forthwith  you  see 

That  all  this  might  otherwise  be. 

Equations,  written  with  pencil  or  pen, 

Must  be  visible  in  space,  and  when 

Difficulties  in  construction  arise, 

We  need  only  define  it  otherwise. 

For,  what  is  formed  after  laws  arithmetic 

Must  also  yield  some  delight  geometric. 

Therefore  we  must  not  object 

That  all  circles  intersect 

In  the  circular  points  at  infinity. 

And  all  parallels,  they  declare, 

If  produced  must  meet  somewhere. 

So  in  space,  it  can't  be  denied, 

Any  old  curvature  may  abide. 

The  formulas  are  all  fine  and  true, 

Then  why  should  they  not  have  a  meaning  too? 

Pupils  everywhere  praise  their  fate 

That  that  now  is  crooked  which  once  was 

straight. 

Non-Euclidean,  in  fine  derision, 
Is  what  it's  called  by  the  geometrician. 
Fuchs.  I  do  not  fully  follow  thee. 
Meph.  No  better  does  philosophy. 

To  master  mathematical  speculation, 

Carefully  learn  to  reduce  your  equation 

By  an  adequate  transformation 

Till  the  formulas  are  devoid  of  interpretation.] 


CHAPTER  XXI 

PARADOXES  AND  CURIOSITIES 

2101.  The  pseudomath  is  a  person  who  handles  mathematics 
as  a  monkey  handles  the  razor.    The  creature  tried  to  shave 
himself  as  he  had  seen  his  master  do;  but,  not  having  any  notion 
of  the  angle  at  which  the  razor  was  to  be  held,  he  cut  his 
own  throat.    He  never  tried  it  a  second  time,  poor  animal !  but 
the  pseudomath  keeps  on  in  his  work,  proclaims  himself  clean 
shaved,  and  all  the  rest  of  the  world  hairy. 

The  graphomath  is  a  person  who,  having  no  mathematics, 
attempts  to  describe  a  mathematician.  Novelists  perform  in 
this  way:  even  Walter  Scott  now  and  then  burns  his  fingers. 
His  dreaming  calculator,  Davy  Ramsay,  swears  "  by  the  bones 
of  the  immortal  Napier."  Scott  thought  that  the  philomaths 
worshipped  relics:  so  they  do  in  one  sense. — DE  MORGAN,  A. 
Budget  of  Paradoxes  (London,  1872),  p.  473. 

2102.  Proof  requires  a  person  who  can  give  and  a  person  who 
can  receive.  .  .  . 

A  blind  man  said,  As  to  the  Sun, 
I'll  take  my  Bible  oath  there's  none ; 
For  if  there  had  been  one  to  show 
They  would  have  shown  it  long  ago. 
How  came  he  such  a  goose  to  be? 
Did  he  not  know  he  couldn't  see? 
Not  he. 

DE  MORGAN,  A. 
Budget  of  Paradoxes  (London,  1872),  p.  262. 

2103.  Mathematical  research,  with  all  its  wealth  of  hidden 
treasure,  is  all  too  apt  to  yield  nothing  to  our  research:  for  it 
is  haunted  by  certain  ignes  fatui — delusive  phantoms,  that 
float  before  us,  and  seem  so  fair,  and  are  all  but  in  our  grasp,  so 
nearly  that  it  never  seems  to  need  more  than  one  step  further, 
and  the  prize  shall  be  ours !    Alas  for  him  who  has  been  turned 

364 


PARADOXES   AND    CURIOSITIES  365 

aside  from  real  research  by  one  of  these  spectres — who  has 
found  a  music  in  its  mocking  laughter — and  who  wastes  his 
life  and  energy  in  the  desperate  chase! — DODGSON,  C.  L. 

A  new  Theory  of  Parallels  (London,  1895), 
Introduction. 

2104.  As  lightning  clears  the  air  of  impalpable  vapours,  so  an 
incisive  paradox  frees  the  human  intelligence  from  the  lethargic 
influence  of  latent  and  unsuspected  assumptions.    Paradox  is 
the  slayer  of  Prejudice. — SYLVESTER,  J.  J. 

On  a  Lady's  Fan  etc.  Collected  Mathematical 
Papers,  Vol.  8,  p.  86. 

2105.  When  a  paradoxer  parades  capital  letters  and  diagrams 
which  are  as  good  as  Newton's  to  all  who  know  nothing  about  it, 
some  persons  wonder  why  science  does  not  rise  and  triturate  the 
whole  thing.    This  is  why :  all  who  are  fit  to  read  the  refutation 
are  satisfied  already,  and  can,  if  they  please,  detect  the  paradoxer 
for  themselves.    Those  who  are  not  fit  to  do  this  would  not  know 
the  difference  between  the  true  answer  and  the  new  capitals  and 
diagrams  on  which  the  delighted  paradoxer  would  declare  that 
he  had  crumbled  the  philosophers,  and  not  they  him. 

DE  MORGAN,  A. 
A  Budget  of  Paradoxes  (London,  1872],  p.  484- 

2106.  Demonstrative  reason  never  raises  the  cry  of  Church  in 
Danger!  and  it  cannot  have  any  Dictionary  of  heresies  except  a 
Budget  of  Paradoxes.    Mistaken  claimants  are  left  to  Time  and 
his  extinguisher,  with  the  approbation  of  all  non-claimants: 
there  is  no  need  of  a  succession  of  exposures.    Time  gets  through 
the  job  in  his  own  workmanlike  manner. — DE  MORGAN,  A. 

A  Budget  of  Paradoxes  (London,  1872},  p.  485. 

2107.  D'Israeli  speaks  of  the  "  six  follies  of  science," — the 
quadrature,  the  duplication,  the  perpetual  motion,  the  philos- 
opher's stone,  magic,  and  astrology.    He  might  as  well  have 
added  the  trisection,  to  make  the  mystic  number  seven;  but 
had  he  done  so,  he  would  still  have  been  very  lenient;  only 
seven  follies  in  all  science,  from  mathematics  to  chemistry! 
Science  might  have  said  to  such  a  judge — as  convicts  used  to 


366  MEMORABILIA   MATHEMATICA 

say  who  got  seven  years,  expecting  it  for  life,  "  Thank  you,  my 
Lord,  and  may  you  sit  there  until  they  are  over," — may  the 
Curiosities  of  Literature  outlive  the  Follies  of  Science! 

DE  MORGAN,  A. 

A  Budget  of  Paradoxes  (London,  1872),  p.  71. 

2108.  Montucla  says,  speaking  of  France,  that  he  finds  three 
notions  prevalent  among  cyclometers:  1.  That  there  is  a  large 
reward  offered  for  success;  2.  That  the  longitude  problem  de- 
pends on  that  success;  3.  That  the  solution  is  the  great  end  and 
object  of  geometry.    The  same  three  notions  are  equally  prev- 
alent among  the  same  class  in  England.     No  reward  has  ever 
been  offered  by  the  government  of  either  country.    The  lon- 
gitude problem  hi  no  way  depends  upon  perfect  solution;  exist- 
ing approximations  are  sufficient  to  a  point  of  accuracy  far 
beyond  what  can  be  wanted.     And  geometry,  content  with 
what  exists,  has  long  passed  on  to  other  matters.    Sometimes  a 
cyclometer  persuades  a  skipper  who  has  made  land  in  the 
wrong  place  that  the  astronomers  are  at  fault,  for  using  a 
wrong  measure  of  the  circle;  and  the  skipper  thinks  it  a  very 
comfortable  solution!    And  this  is  the  utmost  that  the  problem 
has  to  do  with  longitude. — DE  MORGAN,  A. 

A  Budget  of  Paradoxes  (London,  1872),  p.  96. 

2109.  Gregory  St.  Vincent  is  the  greatest  of  circle-squarers, 
and  his  investigations  led  him  into  many  truths:  he  found  the 
property  of  the  arc  of  the  hyperbola  which  led  to  Napier's 
logarithms  being  called  hyperbolic.    Montucla  says  of  him,  with 
sly  truth,  that  no  one  ever  squared  the  circle  with  so  much 
genius,  or,  excepting  his  principal  object,  with  so  much  success. 

DE  MORGAN,  A. 
A  Budget  of  Paradoxes  (London,  1872),  p.  70. 

2110.  When  I  reached  geometry,  and  became  acquainted  with 
the  proposition  the  proof  of  which  has  been  sought  for  centuries, 
I  felt  irresistibly  impelled  to  try  my  powers  at  its  discovery. 
You  will  consider  me  foolish  if  I  confess  that  I  am  still  earnestly 
of  the  opinion  to  have  succeeded  in  my  attempt. 

BOLZANO,  BERNARD. 
Selbstbiographie  (Wien,  187S),  p.  19. 


PARADOXES   AND    CURIOSITIES  367 

2111.  The  Theory  of  Parallels. 

It  is  known  that  to  complete  the  theory  it  is  only  necessary  to 
demonstrate  the  following  proposition,  which  Euclid  assumed  as 
an  axiom: 

Prop.  If  the  sum  of  the  interior  angles  ECF  and  DEC  which 
two  straight  lines  EC  and  DB  make  with  a  third  line  CP  is  less 
than  two  right  angles,  the  lines,  if  sufficiently  produced,  will 
intersect. 


Proof.  Construct  PCA  equal  to  the  supplement  PBD  of 
CBD,  and  ECF,  FCG,  etc.  each  equal  to  ACE,  so  that  ACF  = 
2.ACE,  ACG  =  3.ACE,  etc.  Then  however  small  the  angle 
ACE  may  be,  there  exists  some  number  n  such  that  n.ACE  = 
ACH  will  be  equal  to  or  greater  than  ACP. 

Again,  take  BI,  IL,  etc.  each  equal  to  CB,  and  draw  IK,  LM, 
etc.  parallel  to  BD,  then  the  figures  ACBD,  DBIK,  KILM,  etc. 
are  congruent,  and  ACIK  =  2.ABCD,  ACLM  =  3.ACBD,  etc. 

Take  ACNO  =n.ACBD,  n  having  the  same  value  as  in  the 
expression  ACH  =  n.ACE,  then  ACNO  is  certainly  less  than 
ACP,  since  ACNO  must  be  increased  by  ONP  to  be  equal  to 
ACP.  It  follows  that  ACNO  is  also  less  than  ACH,  and  by 
taking  the  nth  part  of  each  of  these,  that  ACBD  is  less  than 
ACE. 

But  if  ACE  is  greater  than  ACBD,  CE  and  BD  must  inter- 
sect, for  otherwise  ACE  would  be  a  part  of  ACBD. 

Journal  fur  Mathematik,  Bd.  2  (1884),  P- 198. 

2112.  Are  you  sure  that  it  is  impossible  to  trisect  the  angle  by 
Euclid?  I  have  not  to  lament  a  single  hour  thrown  away  on  the 


368  MEMORABILIA   MATHEMATICA 

attempt,  but  fancy  that  it  is  rather  a  tact,  a  feeling,  than  a 
proof,  which  makes  us  think  that  the  thing  cannot  be  done. 
But  would  Gauss's  inscription  of  the  regular  polygon  of  seven- 
teen sides  have  seemed,  a  century  ago,  much  less  an  impossible 
thing,  by  line  and  circle? — HAMILTON,  W.  R. 

Letter  to  De  Morgan  (1852). 

2113.  One  of  the  most  curious  of  these  cases  [geometrical 
paradoxers]  was  that  of  a  student,  I  am  not  sure  but  a  graduate, 
of  the  University  of  Virginia,  who  claimed  that  geometers  were 
in  error  in  assuming  that  a  line  had  no  thickness.    He  published 
a  school  geometry  based  on  his  views,  which  received  the  en- 
dorsement of  a  well-known  New  York  school  official  and,  on  the 
basis  of  this,  was  actually  endorsed,  or  came  very  near  being  en- 
dorsed, as  a  text-book  hi  the  public  schools  of  New  York. 

NEWCOMB,  SIMON. 

The  Reminiscences  of  an  Astronomer  (Boston 
and  New  York,  1903),  p.  388. 

2114.  What  distinguishes  the  straight  line  and  circle  more 
than  anything  else,  and  properly  separates  them  for  the  pur- 
pose of  elementary  geometry?     Their  self-similarity.     Every 
inch  of  a  straight  line  coincides  with  every  other  inch,  and  off  a 
circle  with  every  other  off  the  same  circle.    Where,  then,  did 
Euclid  fail?    In  not  introducing  the  third  curve,  which  has  the 
same  property — the  screw.     The  right  line,   the   circle,   the 
screw — the  representations  of  translation,  rotation,  and  the 
two  combined — ought  to  have  been  the  instruments  of  geom- 
etry.   With  a  screw  we  should  never  have  heard  of  the  impos- 
sibility of  trisecting  an  angle,  squaring  the  circle,  etc. 

DE  MORGAN,  A. 

Quoted  in  Graves'  Life  of  Sir  W.  R.  Hamilton, 
Vol.  3  (New  York,  1889),  p.  342. 

2116.         Mad  Mathesis  alone  was  unconfined, 

Too  mad  for  mere  material  chains  to  bind, 
Now  to  pure  space  lifts  her  ecstatic  stare, 
Now,  running  round  the  circle,  finds  it  square. 

POPE,  ALEXANDER. 
The  Dunciad,  Bk.  4,  lines  31-34- 


PARADOXES   AND   CURIOSITIES  369 

2116.  Or  is't  a  tart  idea,  to  procure 

An  edge,  and  keep  the  practic  soul  in  ure, 

Like  that  dear  Chymic  dust,  or  puzzling  quadrature? 

QUARLES,  PHILIP. 

Quoted  by  De  Morgan:  Budget  of  Paradoxes 
(London,  1872),  p.  436. 

2117.  Quale  e'l  geometra  che  tutto  s'  affige 
Per  misurar  lo  cerchio,  e  non  ritruova, 
Pensando  qual  principio  ond'  egli  indige. — DANTE. 

Paradise,    canto    S3,    lines    122-125. 

[As  doth  the  expert  geometer  appear 

Who  seeks  to  square  the  circle,  and  whose  skill 

Finds  not  the  law  with  which  his  course  to  steer.*] 

Quoted  in  Frankland's  Story  of  Euclid  (London, 

1902),  p.  101. 

2118.  In  Mathematicks  he  was  greater 
Than  Tycho  Brake,  or  Erra  Pater: 
For  he,  by  Geometrick  scale, 
Could  take  the  size  of  Pots  of  Ale; 
Resolve  by  Signs  and  Tangents  streight, 
If  Bread  or  Butter  wanted  weight; 

And  wisely  tell  what  hour  o'  th'  day 
The  Clock  doth  strike,  by  Algebra. 

BUTLER,  SAMUEL. 
Hudibras,  Part  1,  canto  1,  lines  119-126. 

2119.  I  have  often  been  surprised  that  Mathematics,  the 
quintessence  of  truth,  should  have  found  admirers  so  few  and  so 
languid.    Frequent  considerations  and  minute  scrutiny  have  at 
length  unravelled  the  cause;  viz.  that  though  Reason  is  feasted, 
Imagination  is  starved;  whilst  Reason  is  luxuriating  in  its 
proper  Paradise,  Imagination  is  wearily  travelling  on  a  dreary 
desert. — COLERIDGE,  SAMUEL. 

A  Mathematical  Problem. 

2120.  At  last  we  entered  the  palace,  and  proceeded  into  the 
chamber  of  presence  where  I  saw  the  king  seated  on  his  throne, 

*  For  another  rendition  of  these  same  lines  see  1858. 


370  MEMORABILIA   MATHEMATICA 

attended  on  each  side  by  persons  of  prime  quality.  Before  the 
throne,  was  a  large  table  filled  with  globes  and  spheres,  and  math- 
ematical instruments  of  all  kinds.  His  majesty  took  not  the 
least  notice  of  us,  although  our  entrance  was  not  without  suffi- 
cient noise,  by  the  concourse  of  all  persons  belonging  to  the 
court.  But  he  was  then  deep  in  a  problem,  and  we  attended 
an  hour,  before  he  could  solve  it.  There  stood  by  him,  on  each 
side,  a  young  page  with  flaps  in  their  hands,  and  when  they  saw 
he  was  at  leisure,  one  of  them  gently  struck  his  mouth,  and  the 
other  his  right  ear;  at  which  he  started  like  one  awaked  on  the 
sudden,  and  looking  toward  me  and  the  company  I  was  in, 
recollected  the  occasion  of  our  coming,  whereof  he  had  been 
informed  before.  He  spake  some  words,  whereupon  immedi- 
ately a  young  man  with  a  flap  came  to  my  side,  and  flapt  me 
gently  on  the  right  ear,  but  I  made  signs,  as  well  as  I  could, 
that  I  had  no  occasion  for  such  an  instrument;  which,  as  I 
afterwards  found,  gave  his  majesty,  and  the  whole  court,  a  very 
mean  opinion  of  my  understanding.  The  king,  as  far  as  I 
could  conjecture,  asked  me  several  questions,  and  I  addressed 
myself  to  him  in  all  the  languages  I  had.  When  it  was  found, 
that  I  could  neither  understand  nor  be  understood,  I  was  con- 
ducted by  his  order  to  an  apartment  in  his  palace,  (this  prince 
being  distinguished  above  all  his  predecessors,  for  his  hospi- 
tality to  strangers)  where  two  servants  were  appointed  to 
attend  me.  My  dinner  was  brought,  and  four  persons  of 
quality,  did  me  the  honour  to  dine  with  me.  We  had  two 
courses  of  three  dishes  each.  In  the  first  course,  there  was  a 
shoulder  of  mutton  cut  into  an  equilateral  triangle,  a  piece  of 
beef  into  a  rhomboides,  and  a  pudding  into  a  cycloid.  The 
second  course,  was,  two  ducks  trussed  up  in  the  form  of  fiddles; 
sausages  and  puddings,  resembling  flutes  and  haut-boys,  and  a 
breast  of  veal  in  the  shape  of  a  harp.  The  servants  cut  our 
bread  into  cones,  cylinders,  parallelograms,  and  several  other 
mathematical  figures. — SWIFT,  JONATHAN. 

Gulliver's  Travels;  A  Voyage  to  Laputa,  Chap.  2. 

2121.  Those  to  whom  the  king  had  entrusted  me,  observing 
how  ill  I  was  clad,  ordered  a  taylor  to  come  next  morning,  and 
take  measure  for  a  suit  of  cloaths.  This  operator  did  his  office 


PARADOXES   AND    CURIOSITIES  371 

after  a  different  manner,  from  those  of  his  trade  in  Europe.  He 
first  took  my  altitude  by  a  quadrant,  and  then,  with  rule  and 
compasses,  described  the  dimensions  and  outlines  of  my  whole 
body,  all  which  he  entered  upon  paper;  and  in  six  days,  brought 
my  cloaths  very  ill  made,  and  quite  out  of  shape,  by  happening 
to  mistake  a  figure  in  the  calculation.  But  my  comfort  was,  that 
I  observed  such  accidents  very  frequent,  and  little  regarded. 

SWIFT,  JONATHAN. 
Gulliver's  Travels;  A  Voyage  to  Laputa,  Chap.  2. 

2122.  The  knowledge  I  had  in  mathematics,  gave  me  great 
assistance  in  acquiring  their  phraseology,  which  depended  much 
upon  that  science,  and  music;  and  in  the  latter  I  was  not  un- 
skilled.    Their  ideas  are  perpetually  conversant  in  lines  and 
figures.     If  they  would,  for  example,  praise  the  beauty  of  a 
woman,  or  any  other  animal,  they  describe  it  by  rhombs,  circles, 
parallelograms,  ellipses,  and  other  geometrical  terms,  or  by 
words  of  art  drawn  from  music,  needless  here  to  repeat.     I 
observed  in  the  king's  kitchen  all  sorts  of  mathematical  and 
musical  instruments,  after  the  figures  of  which,  they  cut  up  the 
joints  that  were  served  to  his  majesty's  table. 

SWIFT,  JONATHAN. 
Gulliver's  Travels;  A  Voyage  to  Laputa,  Chap.  2. 

2123.  I  was  at  the  mathematical  school,  where  the  master 
taught  his  pupils,  after  a  method,  scarce  imaginable  to  us  in 
Europe.     The  propositions,  and  demonstrations,  were  fairly 
written  on  a  thin  wafer,  with  ink  composed  of  a  cephalic  tincture. 
This,  the  student  was  to  swallow  upon  a  fasting  stomach,  and 
for  three  days  following,  eat  nothing  but  bread  and  water.    As 
the  wafer  digested,  the  tincture  mounted  to  his  brain,  bearing 
the  proposition  along  with  it.    But  the  success  has  not  hitherto 
been  answerable,  partly  by  some  error  in  the  quantum  or  com- 
position, and  partly  by  the  perverseness  of  lads;  to  whom  this 
bolus  is  so  nauseous,  that  they  generally  steal  aside,  and  dis- 
charge it  upwards,  before  it  can  operate;  neither  have  they  been 
yet  persuaded  to  use  so  long  an  abstinence  as  the  prescription 
requires. — SWIFT,  JONATHAN. 

Gulliver's  Travels;  A  Voyage  to  Laputa,  Chap.  5. 


372  MEMORABILIA   MATHEMATICA 

2124.  It  is  worth  observing  that  some  of  those  who  disparage 
some  branch  of  study  in  which  they  are  deficient,  will  often 
affect  more  contempt  for  it  than  they  really  feel.    And  not  un- 
frequently  they  will  take  pains  to  have  it  thought  that  they  are 
themselves  well  versed  hi  it,  or  that  they  easily  might  be,  if  they 
thought  it  worth  while; — in  short,  that  it  is  not  from  hanging  too 
high  that  the  grapes  are  called  sour. 

Thus,  Swift,  in  the  person  of  Gulliver,  represents  himself, 
while  deriding  the  extravagant  passion  for  Mathematics  among 
the  Laputians,  as  being  a  good  mathematician.  Yet  he  betrays 
his  utter  ignorance,  by  speaking  "  of  a  pudding  in  the  form  of  a 
cycloid: "  evidently  taking  the  cycloid  for  a  figure,  instead  of  a 
line.  This  may  help  to  explain  the  difficulty  he  is  said  to  have 
had  in  obtaining  his  Degree. — WHATELY,  R. 

Annotations   to   Bacon's   Essays,    Essay    L. 

2125.  It  is  natural  to  think  that  an  abstract  science  cannot  be 
of  much  importance  in  the  affairs  of  human  life,  because  it  has 
omitted  from  its  consideration  everything  of  real  interest.    It 
will  be  remembered  that  Swift,  hi  his  description  of  Gulliver's 
voyage  to  Laputa,  is  of  two  minds  on  this  point.    He  describes 
the  mathematicians  of  that  country  as  silly  and  useless  dreamers, 
whose  attention  has  to  be  awakened  by  flappers.    Also,  the 
mathematical  tailor  measures  his  height  by  a  quadrant,  and 
deduces  his  other  dimensions  by  a  rule  and  compasses,  producing 
a  suit  of  very  ill-fitting  clothes.    On  the  other  hand,  the  mathe- 
maticians of  Laputa,  by  then*  marvellous  invention  of  the 
magnetic  island  floating  hi  the  air,  ruled  the  country  and  main- 
tained their  ascendency  over  their  subjects.     Swift,  indeed, 
lived  at  a  time  peculiarly  unsuited  for  gibes  at  contemporary 
mathematicians.     Newton's  Prindpia  had  just  been  written, 
one  of  the  great  forces  which  have  transformed  the  modern 
world.    Swift  might  just  as  well  have  laughed  at  an  earthquake. 

WHITEHEAD,  A.  N. 

An  Introduction  to  Mathematics  (New  York, 
1911),  p.  10. 


PARADOXES   AND   CURIOSITIES 


373 


Here  I  am  as  yon  may  «w 

a«  4  b'  -  ab 
k  When  two  Triangle*  on  me  stand 
/  ''\8quare  of  hypothec'  a  plann'd 


.  But  if  I  stand  on  than  instead, 


^m.^^iiiiiii 


AIRY,  G.  B. 

Quoted  in  Graves1  Life  of  Sir  W.  R.  Hamilton, 
Vol.  8  (New  York,  1889),  p.  502. 

2127.  TT"=  3.141  592  653  589  793  238  462  643  383  279  . 

31415  9 

Now  I,  even  I,  would  celebrate 
26  535 

In  rhymes  inapt,  the  great 

8979 
Immortal  Syracusan,  rivaled  nevermore, 

323          8  4 

Who  in  his  wondrous  lore, 

626 
Passed  on  before, 

433          8          327  9 

Left  men  his  guidance  how  to  circles  mensurate. 

ORR,  A.  C. 
Literary  Digest,   Vol.  32  (1906),  p.  84. 

2128.  I  take  from  a  biographical  dictionary  the  first  five 
names  of  poets,  with  their  ages  at  death.    They  are 

Aagard,  died  at  48. 
Abeille,  "  "  76. 
Abulola,  "  "  84. 
Abunowas,  "  "  48. 
Accords,  "  "  45. 
These  five  ages  have  the  following  characters  in  common: — 


374  MEMORABILIA   MATHEMATICA 

1.  The  difference  of  the  two  digits  composing  the  number, 
divided  by  three,  leaves  a  remainder  of  one. 

2.  The  first  digit  raised  to  the  power  indicated  by  the  second, 
and  then  divided  by  three,  leaves  a  remainder  of  one. 

3.  The  sum  of  the  prime  factors  of  each  age,  including  one  as  a 
prime  factor,  is  divisible  by  three. — PEIRCE,  C.  S. 

A   Theory  of  Probable  Inference;  Studies  in 
Logic  (Boston,  1888),  p.  163. 

2129.  In  view  of  the  fact  that  the  offered  prize  [for  the  solu- 
tion of  the  problem  of  Fermat's  Greater  Theorem]  is  about 
$25,000  and  that  lack  of  marginal  space  hi  his  copy  of  Diophan- 
tus  was  the  reason  given  by  Fermat  for  not  communicating  his 
proof,  one  might  be  tempted  to  wish  that  one  could  send  credit 
tor  a  dime  back  through  the  ages  to  Fermat  and  thus  secure  this 
coveted  prize,  if  it  actually  existed.    This  might,  however,  result 
more  seriously  than  one  would  at  first  suppose;  for  if  Fermat  had 
bought  on  credit  a  dime's  worth  of  paper  even  during  the  year  of 
his  death,  1665,  and  if  this  bill  had  been  drawing  compound 
interest  at  the  rate  of  six  per  cent,  since  that  time,  the  bill  would 
now  amount  to  more  than  seven  times  as  much  as  the  prize. 

MILLER,  G.  A. 

Some    Thoughts    on    Modern    Mathematical 
Research;  Science,    Vol.  35   (1912],  p.  881. 

2130.  //  the  Indians  hadn't  spent  the  $24.     In  1626  Peter 
Minuit,  first  governor  of  New  Netherland,  purchased  Manhattan 
Island  from  the  Indians  for  about  $24.    The  rate  of  interest  on 
money  is  higher  in  new  countries,  and  gradually  decreases  as 
wealth  accumulates.    Within  the  present  generation  the  legal 
rate  in  the  state  has  fallen  from  7%  to  6%.    Assume  for  sim- 
plicity a  uniform  rate  of  7%  from  1626  to  the  present,  and 
suppose  that  the  Indians  had  put  their  $24  at  interest  at  that 
rate  (banking  facilities  in  New  York  being  always  taken  for 
granted!)  and  had  added  the  interest  to  the  principal  yearly. 
What  would  be  the  amount  now,  after  280  years?    24  x  ( 1 .07)  28° 

=more  than  4,042,000,000. 

The  latest  tax  assessment  available  at  the  time  of  writing  gives 
the  realty  for  the  borough  of  Manhattan  as  $3,820,754.181. 


PARADOXES   AND    CURIOSITIES  375 

This  is  estimated  to  be  78%  of  the  actual  value,  making  the 
actual  value  a  little  more  than  $4,898,400,000. 

The  amount  of  the  Indians'  money  would  therefore  be  more 
than  the  present  assessed  valuation  but  less  than  the  actual 
valuation. — WHITE,  W.  F. 

A    Scrap-book    of    Elementary    Mathematics 

(Chicago,  1908),  pp.  47-48. 

2131.  See  Mystery  to  Mathematics  fly! — POPE,  ALEXANDER. 

The  Dunciad,  Bk.  4,  line  647. 

2132.  The  Pythagoreans  and  Platonists  were  carried  further 
by  this  love  of  simplicity.    Pythagoras,  by  his  skill  in  mathe- 
matics, discovered  that  there  can  be  no  more  than  five  regular 
solid  figures,  terminated  by  plane  surfaces  which  are  all  similar 
and  equal;  to  wit,  the  tetrahedron,  the  cube,  the  octahedron,  the 
dodecahedron,  and  the  eicosihedron.    As  nature  works  in  the 
most  simple  and  regular  way,  he  thought  that  all  elementary 
bodies  must  have  one  or  other  of  those  regular  figures;  and  that 
the  discovery  of  the  properties  and  relations  "of  the  regular  solids 
must  be  a  key  to  open  the  mysteries  of  nature. 

This  notion  of  the  Pythagoreans  and  Platonists  has  undoubt- 
edly great  beauty  and  simplicity.  Accordingly  it  prevailed,  at 
least  to  the  time  of  Euclid.  He  was  a  Platonic  philosopher,  and 
is  said  to  have  wrote  all  the  books  of  his  Elements,  in  order  to 
discover  the  properties  and  relations  of  the  five  regular  solids. 
The  ancient  tradition  of  the  intention  of  Euclid  in  writing  his 
elements,  is  countenanced  by  the  work  itself.  For  the  last  book 
of  the  elements  treats  of  the  regular  solids,  and  all  the  preceding 

are  subservient  to  the  last. — REID,  THOMAS. 

Essays  on  the  Powers  of  the  Human  Mind 
(Edinburgh,  1812),  Vol.  2,  p.  400. 

2133.  In  the  Timaeus  [of  Plato]  it  is  asserted  that  the  parti- 
cles of  the  various  elements  have  the  forms  of  these  [the  regular] 
solids.    Fire  has  the  Pyramid;  Earth  has  the  Cube;  Water  the 
Octahedron;  Air  the  Icosahedron;  and  the  Dodecahedron  is  the 
plan  of  the  Universe  itself.    It  was  natural  that  when  Plato  had 
learnt  that  other  mathematical  properties  had  a  bearing  upon 
the  constitution  of  the  Universe,  he  should  suppose  that  the 


376  MEMORABILIA   MATHEMATICA 

singular  property  of  space,  which  the  existence  of  this  limited 
and  varied  class  of  solids  implied,  should  have  some  correspond- 
ing property  in  the  Universe,  which  exists  in  space. 

WHEWELL,  W. 

History  of  the  Inductive  Sciences,  3rd  Edition, 
Additions  to  Bk.  2. 

2134.  The  orbit  of  the  earth  is  a  circle:  round  the  sphere  to 
which  this  circle  belongs,  describe  a  dodecahedron;  the  sphere 
including  this  will  give  the  orbit  of  Mars.     Round  Mars  de- 
scribe a  tetrahedron;  the  circle  including  this  will  be  the  orbit  of 
Jupiter.     Describe  a  cube  round  Jupiter's  orbit;  the  circle 
including  this  will  be  the  orbit  of  Saturn.    Now  inscribe  in  the 
earth's  orbit  an  icosahedron;  the  circle  inscribed  in  it  will  be 
the  orbit  of  Venus.     Inscribe  an  octahedron  in  the  orbit  of 
Venus;  the  circle  inscribed  hi  it  will  be  Mercury's  orbit.    This 
is  the  reason  of  the  number  of  the  planets. — KEPLER. 

Mysterium    Cosmographicum    [Whewell]. 

2135.  It  will  not  be  thought  surprising  that  Plato  expected 
that  Astronomy,  when  further  advanced,  would  be  able  to 
render  an  account  of  many  things  for  which  she  has  not  ac- 
counted even  to  this  day.    Thus,  in  the  passage  hi  the  seventh 
Book  of  the  Republic,  he  says  that  the  philosopher  requires  a 
reason  for  the  proportion  of  the  day  to  the  month,  and  the  month 
to  the  year,  deeper  and  more  substantial  than  mere  observation 
can  give.    Yet  Astronomy  has  not  yet  shown  us  any  reason  why 
the  proportion  of  the  tunes  of  the  earth's  rotation  on  its  axis,  the 
moon's  revolution  round  the  earth,  and  the  earth's  revolution 
round  the  sun,  might  not  have  been  made  by  the  Creator  quite 
different  from  what  they  are.     But  hi  asking  Mathematical 
Astronomy  for  reasons  which  she  cannot  give,  Plato  was  only 
doing  what  a  great  astronomical  discoverer,  Kepler,  did  at  a 
later  period.     One  of  the  questions  which  Kepler  especially 
wished  to  have  answered  was,  why  there  are  five  planets,  and 
why  at  such  particular  distances  from  the  sun?    And  it  is  still 
more  curious  that  he  thought  he  had  found  the  reason  of  these 
things,  in  the  relation  of  those  five  regular  solids  which  Plato  was 
desirous  of  introducing  into  the  philosophy  of  the  universe.  .  .  . 


PARADOXES   AND    CURIOSITIES  377 

Kepler  regards  the  law  which  thus  determines  the  number 
and  magnitude  of  the  planetary  orbits  by  means  of  the  five 
regular  solids  as  a  discovery  no  less  remarkable  and  certain 
than  the  Three  Laws  which  give  his  name  its  imperishable  place 
in  the  history  of  astromomy. — WHEWELL,  W. 

History  of  the  Inductive  Sciences,  3rd  Edition, 
Additions  to  Bk.  8. 

2136.  Pythagorean    philosophers  .  .  .  maintained    that    of 
two  combatants,  he  would  conquer,  the  sum  of  the  numbers  ex- 
pressed by  the  characters  of  whose  names  exceeded  the  sum  of 
those  expressed  by  the  other.    It  was  upon  this  principle  that 
they  explained  the  relative  prowess  and  fate  of  the  heroes  in 
Homer,  Harpotc\o<;)   'E/cTco/a   and  A^tXXeu?,   the  sum  of  the 
numbers  hi  whose  names  are  861,  1225,  and  1276  respectively. 

PEACOCK,  GEORGE. 

Encyclopedia  of  Pure  Mathematics  (London, 
1847);  Article  "Arithmetic,"  sect.  88. 

2137.  Round  numbers  are  always  false. — JOHNSON,  SAMUEL. 

Johnsoniana;   Apothegms,    Sentiment,    etc. 

2138.  Numero  deus  impare  gaudet  [God  in  number  odd  re- 
joices.]— VIRGIL. 

Eclogue,  8,   77. 

2139.  Why  is  it  that  we  entertain  the  belief  that  for  every 
purpose  odd  numbers  are  the  most  effectual? — PLINY. 

Natural    History,    Bk.    28,    chap.    5. 

2140.  "Then  here  goes  another,"  says   he,  "to  make  sure, 
Fore  there's  luck  in  odd  numbers,"  says  Rory  O'Moore. 

LOVER,  S. 
Rory  O'Moore. 

2141.  This  is  the  third  time;  I  hope,  good  luck  lies  in  odd 
numbers.  .  .  .  They  say,  there  is  divinity  in  odd  numbers, 
either  in  nativity,  chance,  or  death. — SHAKESPEARE. 

The  Merry  Wives  of  Windsor,  Act  5,  scene  1. 


378  MEMORABILIA   MATHEMATICA 

2142.  To  add  to  golden  numbers,  golden  numbers. 

DECKER,  THOMAS. 
Patient  Grissell,  Act  1,  scene  1. 

2143.  I've  read  that  things  inanimate  have  moved, 
And,  as  with  living  souls,  have  been  inform'd, 
By  magic  numbers  and  persuasive  sound. 

CONGREVE,  RICHARD. 

The  Morning  Bride,  Act  1,  scene  1. 

2144.  .  .  .  the  Yancos  on  the  Amazon,   whose  name  for 
three  is 

Poettarrarorincoaroac, 

of  a  length  sufficiently  formidable  to  justify  the  remark  of 
La  Condamine:  Heureusement  pour  ceux  qui  ont  a  faire  avec 
eux,  leur  Arithmetique  ne  va  pas  plus  loin. — PEACOCK,  GEORGE. 

Encyclopedia  of  Pure  Mathematics  (London, 
1847);  Article  "Arithmetic,"  sect.  32. 

2145.  There  are  three  principal  sins,  avarice,  luxury,  and 
pride;  three  sorts  of  satisfaction  for  sin,  fasting,  almsgiving,  and 
prayer;  three  persons  offended  by  sin,  God,  the  sinner  himself, 
and  his  neighbour;  three  witnesses  in  heaven,  Pater,  verbum,  and 
spiritus  sanctus;  three  degrees  of  penitence,  contrition,  confes- 
sion, and  satisfaction,  which  Dante  has  represented  as  the 
three  steps  of  the  ladder  that  lead  to  purgatory,  the  first  marble, 
the  second  black  and  rugged  stone,  and  the  third  red  porphyry. 
There  are  three  sacred  orders  in  the  church  militant,  sub- 
diaconati,  diaconiti,  and  presbyterati;  there  are  three  parts,  not 
without  mystery,  of  the  most  sacred  body  made  by  the  priest 
in  the  mass;  and  three  times  he  says  Agnus  Dei,  and  three  times, 
Sanctus;  and  if  we  well  consider  all  the  devout  acts  of  Christian 
worship,  they  are  found  in  a  ternary  combination;  if  we  wish 
rightly  to  partake  of  the  holy  communion,  we  must  three  times 
express  our  contrition,  Domine  non  sum  dignus;  but  who  can 
say  more  of  the  ternary  number  in  a  shorter  compass,  than  what 
the  prophet  says,  tu  signaculum  sanctae  trinitatis.     There  are 
three  Furies  in  the  infernal  regions;  three  Fates,  Atropos,  Lach- 
esis,  and  Clotho.     There  are  three  theological  virtues:  Fides, 


PARADOXES   AND    CURIOSITIES  379 

spes,  and  charitas.  Tria  sunt  pericula  mundi:  Equum  currere; 
navigare,  et  sub  tyranno  vivere.  There  are  three  enemies  of  the 
soul :  the  Devil,  the  world,  and  the  flesh.  There  are  three  things 
which  are  of  no  esteem :  the  strength  of  a  porter,  the  advice  of  a 
poor  man,  and  the  beauty  of  a  beautiful  woman.  There  are 
three  vows  of  the  Minorite  Friars:  poverty,  obedience,  and 
chastity.  There  are  three  terms  in  a  continued  proportion. 
There  are  three  ways  in  which  we  may  commit  sin :  corde,  ore,  ope. 
Three  principal  things  in  Paradise:  glory,  riches,  and  justice. 
There  are  three  things  which  are  especially  displeasing  to  God : 
an  avaricious  rich  man,  a  proud  poor  man,  and  a  luxurious  old 
man.  And  all  things,  in  short,  are  founded  in  three;  that  is,  in 
number,  in  weight,  and  in  measure. 

PACIOLI,  Author  of  the  first  printed  treatise  on  arithmetic. 

Quoted  in  Encyclopedia  of  Pure  Mathematics 
(London,  1847);  Article  "Arithmetic,"  sect.  90. 

2146.  Ah!  why,  ye  Gods,  should  two  and  two  make  four? 

POPE,  ALEXANDER. 

The  Dunciad,  Bk.  2,  line  285. 

2147.  By  him  who  stampt  The  Four  upon  the  mind,— 
The  Four,  the  fount  of  nature's  endless  stream. 

Ascribed  to  PYTHAGORAS. 

Quoted  in  Whewell's  History  of  the  Inductive 
Sciences,  Bk.  4,  chap.  8. 

2148.  Along  the  skiey  arch  the  goddess  trode, 
And  sought  Harmonia's  august  abode; 
The  universal  plan,  the  mystic  Four, 
Defines  the  figure  of  the  palace  floor. 
Solid  and  square  the  ancient  fabric  stands, 
Raised  by  the  labors  of  unnumbered  hands. 

NONNUS. 
Dionysiac,  41,  275-280.    [Whewell]. 

2149.  The  number  seventy-seven  figures  the  abolition  of  all 
sins  by  baptism.  .  .  .  The  number  ten  signifies  justice  and 
beatitude,  resulting  from  the  creature,  which  makes  seven  with 
the  Trinity ,  which  is  three :  therefore  it  is  that  God's  command- 


380  MEMORABILIA   MATHEMATICA 

ments  are  ten  in  number.  The  number  eleven  denotes  sin, 
because  it  transgresses  ten.  .  .  .  This  number  seventy-seven  is 
the  product  of  eleven,  figuring  sin,  multiplied  by  seven,  and  not 
by  ten,  for  seven  is  the  number  of  the  creature.  Three  repre- 
sents the  soul,  which  is  in  some  sort  an  image  of  Divinity;  and 
four  represents  the  body,  on  account  of  its  four  qualities.  .  .  . 

ST.  AUGUSTINE. 
Sermon  41,  art.  23. 

2150.  Heliodorus  says  that  the  Nile  is  nothing  else  than  the 
year,  founding  his  opinion  on  the  fact  that  the  numbers  ex- 
pressed by  the  letters  NetXo?,  Nile,  are  in  Greek  arithmetic, 
N  =  50;  E  =  5;  I  =  10;  A  =  30;  O  =  70;  2  =  200;  and  these  figures 
make  up  together  365,  the  number  of  days  in  the  year. 

Littell's  Living  Age,   Vol.   117,  p.  380. 

2151.  In   treating   666,    Bungus    [Petri    Bungi   Bergomatis 
Numerorum  mysteria,  Bergamo,  1591]  a  good  Catholic,  could 
not  compliment  the  Pope  with  it,  but  he  fixes  it  on  Martin 
Luther  with  a  little  forcing.     If  from  A  to  I  represent  1-10, 
from  K  to  S  10-90,  and  from  T  to  Z  100-500,  we  see— 

MARTIN          LU       TERA 
30     1     80     100    9    40         20    200     100    5    80     1 

which  gives  666.  Again  in  Hebrew,  Lulter  [Hebraized  form  of 
Luther]  does  the  same : — 

n        n      h      i      *> 

200    400    30    6    30 

DE  MORGAN,  A. 

Budget  of  Paradoxes  (London,  1872),  p.  37. 

2152.  Stifel,  the  most  acute  and  original  of  the  early  mathe- 
maticians of  Germany,  .  .  .  relates  .  .  .  that  whilst  a  monk 
at  Esslingen  in  1520,  and  when  infected  by  the  writings  of 
Luther,  he  was  reading  in  the  library  of  his  convent  the  13th 
Chapter  of  Revelations,  it  struck  his  mind  that  the  Beast  must 
signify  the  Pope,  Leo  X.;    He  then  proceeded  in  pious  hope  to 
make  the  calculation  of  the  sum  of  the  numeral  letters  in  Leo 
detimus,  which  he  found  to  be  M,  D,  C,  L,  V,  I;  the  sum  which 
these  formed  was  too  great  by  M,  and  too  little  by  X;  but  he 


PARADOXES   AND    CURIOSITIES  381 

bethought  him  again,  that  he  has  seen  the  name  written  Leo  X., 
and  that  there  were  ten  letters  hi  Leo  decimus,  from  either  of 
which  he  could  obtain  the  deficient  number,  and  by  interpreting 
the  M  to  mean  mysterium,  he  found  the  number  required,  a 
discovery  which  gave  him  such  unspeakable  comfort,  that  he 
believed  that  his  interpretation  must  have  been  an  immediate 
inspiration  of  God. — PEACOCK,  GEORGE. 

Encyclopedia  of  Pure  Mathematics  (London, 
1847};  Article  "Arithmetic,"  sect.  89. 

2153.  Perhaps  the  best  anagram  ever  made  is  that  by  Dr. 
Burney  on  Horatio  Nelson,  so  happily  transformed  into  the  Latin 
sentence  so  truthful  of  the  great  admiral,  Honor  est  a  Nilo. 
Reading  this,  one  is  almost  persuaded  that  the  hit  contained  in 
it  has  a  meaning  provided  by  providence  or  fate. 

This  is  also  amusingly  illustrated  in  the  case  of  the  French- 
man Andre"  Pujom,  who,  using  j  as  i,  found  in  his  name  the 
anagram,  Pendu  a  Riom.  Riom  being  the  seat  of  justice  for  the 
province  of  Auvergne,  the  poor  fellow,  impelled  by  a  sort  of 
infatuation,  actually  committed  a  capital  offence  in  that  pro- 
vince, and  was  hanged  at  Riom,  that  the  anagram  might  be  ful- 
filled. 

New  American  Cyclopedia,   Vol.  1;  Article 
"  Anagram." 

2154.  The  most  remarkable  pseudonym  [of  transposed  names 
adopted  by  authors]  is  the  name  of  "Voltaire,"  which  the 
celebrated  philosopher  assumed  instead  of  his  family  name, 
"Francois  Marie  Arouet,"  and  which  is  now  generally  allowed 
to  be  an  anagram  of  "Arouet,  1.  j.,"  that  is,  Arouet  the  younger. 

Encyclopedia  Britannica,  llth  Edition;  Article 
"Anagram." 

2155.  Perhaps  the  most  beautiful  anagram  that  has  ever 
been  composed  is  by  Jablonsky,  a  former  rector  of  the  school  at 
Lissa.    The  occasion  was  the  following :  When  while  a  young  man 
king  Stanislaus  of  Poland  returned  from  a  journey,  the  whole 
house  of  Lescinsky  assembled  to  welcome  the  family  heir.    On 
this  occasion  Jablonsky  arranged  for  a  school  program,  the 
closing  number  of  which  consisted  of  a  ballet  by  thirteen  pupils 


382  MEMORABILIA   MATHEMATICA 

impersonating  youthful  heroes.  Each  of  them  carried  a  shield 
on  which  appeared  in  gold  one  of  the  letters  of  the  words  Domus 
Lescinia.  At  the  end  of  the  first  dance  the  children  were  so 
arranged  that  the  letters  on  their  shields  spelled  the  words 
Domus  Lescinia.  At  the  end  of  the  second  dance  they  read: 
ades  incolumis  (sound  thou  art  here).  After  the  third:  omnis 
es  lucida  (wholly  brilliant  art  thou) ;  after  the  fourth :  lucida  sis 
omen  (bright  be  the  omen).  Then:  mane  sidus  loci  (remain  our 
country's  star) ;  and  again :  sis  columna  Dei  (be  a  column  of  God) ; 
and  finally:  //  scande  solium  (Proceed,  ascend  the  throne).  This 
last  was  the  more  beautiful  since  it  proved  a  true  prophecy. 
Even  more  artificial  are  the  anagrams  which  transform  one 
verse  into  another.  Thus  an  Italian  scholar  beheld  in  a  dream 
the  line  from  Horace:  Grata  superveniet,  quae  non  sperabitur, 
hora.  This  a  friend  changed  to  the  anagram :  Est  ventura  Rhosina 
parataque  nubere  pigro.  This  induced  the  scholar,  though  an 
old  man,  to  marry  an  unknown  lady  by  the  name  of  Rosina. 

HEIS,  EDUARD. 
Algebraische  Aufgaben  (Koln,  1898),  p.  831. 

2156.  The  following  verses  read  the  same  whether  read  for- 
ward or  backward : — 

Aspice!  nam  raro  mittit  timor  arma,  nee  ipsa 

Si  se  mente  reget,  non  tegeret  Nemesis;  * 
also,  Sator  Arepo  tenet  opera  rotas. 

HEIS,    EDUARD. 
Algebraische  Aufgaben  (Koln,  1898),  p.  828. 

2157.  There  is  a  certain  spiral  of  a  peculiar  form  on  which  a 
point  may  have  been  approaching  for  centuries  the  center,  and 
have  nearly  reached  it,  before  we  discover  that  its  rate  of  ap- 
proach is  accelerated.     The  first  thought  of  the  observer,  on 
seeing  the  acceleration,  would  be  to  say  that  it  would  reach  the 
center  sooner  than  he  had  before  supposed.    But  as  the  point 
comes  near  the  center  it  suddenly,  although  still  moving  under 
the  same  simple  law  as  from  the  beginning,  makes  a  very  short 
turn  upon  its  path  and  flies  off  rapidly  almost  in  a  straight  line, 

*  The  beginning  of  a  poem  which  Johannes  a  Lasco  wrote  on  the 
count  Karl  von  Siidermanland. 


PARADOXES   AND    CURIOSITIES  383 

out  to  an  infinite  distance.  This  illustrates  that  apparent  breach 
of  continuity  which  we  sometimes  find  hi  a  natural  law;  that 
apparently  sudden  change  of  character  which  we  sometimes  see 
in  man. — HILL,  THOMAS. 

Uses  of  Mathesis;  Bibliotheca  Sacra,  Vol.  82, 
p.  521. 

2158.  One  of  the  most  remarkable  of  Babbage's  illustrations 
of  miracles  has  never  had  the  consideration  hi  the  popular  mind 
which  it  deserves;  the  illustration  drawn  from  the  existence  of 
isolated  points  fulfilling  the  equation  of  a  curve.  .  .  .  There  are 
definitions  of  curves  which  describe  not  only  the  positions  of 
every  point  in  a  certain  curve,  but  also  of  one  or  more  perfectly 
isolated  points ;  and  if  we  should  attempt  to  get  by  induction  the 
definition,  from  the  observation  of  the  points  on  the  curve,  we 
might  fail  altogether  to  include  these  isolated  points;  which, 
nevertheless,  although  standing  alone,  as  miracles  to  the 
observer  of  the  course  of  the  points  in  the  curve,  are  nevertheless 
rigorously  included  in  the  law  of  the  curve. — HILL,  THOMAS. 

Uses  of  Mathesis;  Bibliotheca  Sacra,  Vol.  32, 

p.  516. 

2169.  Pure  mathematics  is  the  magician's  real  wand. 

NOVALIS. 

Schriften,  Zweiter  Teil  (Berlin,  1901),  p.  223. 

2160.  Miracles,  considered  as  antinatural  facts,  are  amathe- 
matical,  but  there  are  no  miracles  in  this  sense,  and  those  so 
called  may  be  comprehended  by  means  of  mathematics,  for  to 

mathematics  nothing  is  miraculous. — NOVALIS. 

Schriften,  Zweiter  Teil  (Berlin,  1911),  p.  222. 


INDEX 


Black-faced  numbers  refer  to  authors 

Abbreviations: — m.  =  mathematics,   math.  =  mathematical, 
math'n.  =  mathematician. 


Abbott,  1001. 

Abstract   method,    Development 

of,  729. 
Abstract  nature  of  m.,   Reason 

for,  638. 
Abstractness,   math.,    Compared 

with  logical,  1304. 
Abstract  reasoning,  Objection  to, 

1941. 
Adams,  Henry,  M.  and  history, 

1599. 
Math'ns  practice  freedom,  208, 

805. 

Adams,  John,  Method  in  m.,  226. 
Aeneid,    Euler's    knowledge    of, 

859. 

Aeschylus.    On  number,  1606. 
Aim  in  teaching  m.,  501-508,  517, 

844. 
Airy,     Pythagorean    theorem, 

2126. 

Akenside,  1532. 
Alexander,  901,  902. 
Algebra,  Chapter  XVII. 

Definitions  of,  110,  1714,  1715. 
Problems  in,  320,  530,  1738. 
Of  use  to  grown  men,  425. 
And  geometry,  525-527,  1610, 

1707. 
Advantages    of,     1701,    1703, 

1705. 

Laws  of,  1708-1710. 
As  an  art,  1711. 
Review  of,  1713. 
Designations  of,  1717. 
Origin    of,     1736. 
Burlesque  on  modern,  1741. 
Hume  on,  1863. 

Algebraic    notation,     value    of, 
1213,  1214. 


Algebraic  treatises,  How  to  read, 

601. 

Amusements  hi  m.,  904,  905. 
Anagrams,  On  De  Morgan,  947. 
On  Domus  Lescinia, 

2155. 

On  Flamsteed,  968. 
On  Macaulay,  996. 
On  Nelson,  2153. 
On  Newton,  1028. 
On  Voltaire,  2154. 
Analysis,  Invigorates  the  faculty 

of  resolution,  416. 
Relation  of  geometry  to,  1931. 
Analytical  geometry,  1889,  1890, 

1893. 

Method  of,  310. 
Importance  of,  949. 
Burlesque  on,  2040. 
Ancient  geometry, 

Characteristics    of,    712,    714. 
Compared  with  modern,  1711- 

1716. 

Method  of,  1425,  1873-1875. 
Ancients,  M.  among  the,  321. 
Anecdotes,  Chapters,  IX,  X. 
Anger,    M.    destroys   predisposi- 
tion to,  458. 
Angling  like  m.,  739. 
Anglo-Danes,    Aptitude   for   m., 

836. 
Anglo-Saxons,  Aptitude  for  m., 

837. 
Newton  as  representative  of, 

1014. 
Anonymous,  Song  of  the  screw, 

1894. 

Appolonius,  712,  714. 
Approximate  m.,  Why  not  suffi- 
cient, 1518. 


385 


386 


INDEX 
Black-faced  numbers  refer  to  authors 


Aptitude  for  m.,  509,  510,  520, 

836-838,  976,  1617. 
Arabic  notation,  1614. 
Arago,  M.  the  enemy  of  scientific 

romances,  267. 
Euler,     "analysis    incarnate," 

961. 

Euler  as  a  computer,  962. 
On    Kepler's    discovery,    982. 
Newton's  efforts  superhuman, 

1006. 

On  probabilities,  1691. 
Geometry    as   an   instrument, 

1868. 

Arbuthnot,  M.  frees  from  preju- 
dice, credulity  and  supersti- 
tion, 449. 

M.  the  friend  of  religion,  458. 
M.  compared  to  music,  1112. 
On  math,  reasoning,  1603. 
Archimedes,  His  machines,  903, 

904. 
Estimate  of  math,  appliances, 

904-906,  908. 
Wordsworth  on,  906. 
Schiller  on,  907. 
And  engineering,  908. 
Death  of,  909. 
His  tomb,  910. 

Compared  with  Newton,  911. 
Character  of  his  work,  912,  913. 
Applied  m.,  1312. 
Architecture  and  m.,  276. 
Archytas,  904. 

And  Plato,  1427. 
Aristippus  the  Cyrenaic,  845. 
Aristotle,  914. 

On  relation  of  m.  to  esthetics, 

318. 

Arithmetical  theorems,  1639. 
Arithmetic,  Chapter  XVI. 

Definitions  of,  106,  110,  1611, 

1612,  1714. 
Emerson  on  advantage  of  study 

of,  408. 

Problems  in,  528. 
A  master-key,  1571. 
Based  on  concept  of  time,  1613. 
Method  of  teaching,  1618. 
Purpose  of  teaching,  454,  1624. 
As  logic,  1624,  1625. 


The  queen  of  m.,  1642. 

Higher,  1755. 

Hume  on,  1863. 

Art,  M.  as  a  fine,  Chapter  XI 
Arts,  M.and  the,  1568-1570, 1573. 
Astronomy  and  m.,  1554,  1559, 

1562-1567. 

"Auge  et  impera.,"  631. 
Authority  in  science,  1528. 
Axioms,  518,  2015. 

In  geometry,  1812,  2004,  2006. 

Def .  in  disguise,  2005. 

Euclid's,  2007-2010,  2014. 

Nature  of,  2012. 

Proofs  of,  2013. 

And  the  idea  of  space,  2004. 

Babbage,  923. 
Bacon,  Lord, 
Classification  of  m.,  106. 
M.  makes  men  subtile,  248. 
View  of  m.,  316,  915,  916. 
M.    held   in   high   esteem   by 

ancients,  321. 
On  the  generalizing  power  of 

m.,  327. 
On  the  value  of  math,  studies, 

410. 
M.  develops  concentration  of 

mind,  411. 
M.  cures  distraction  of  mind, 

412. 
M.  essential  to  study  of  nature, 

436. 

His  view  of  m.,  915,  916. 
His  knowledge  of  m.,  917,  918. 
M.  and  logic,  1310. 
Growth  of  m.,  1611. 
Bacon,    Roger,    Neglect    of    m. 

works  injury  to  all  science, 

310. 

On  the  value  of  m.,  1547. 
Bain,  Importance  of  m.  in  educa- 
tion, 442. 
On  the  charm  of  the  study  of 

m.,  453. 

M.  and  science  teaching,  622. 
Teaching  of  arithmetic,   1618. 
Ball,  R.  S.,  2010. 

Ball,  W.  W.  R.,  On  Babbage,  923. 
On  Demoivre's  death,  944. 


INDEX 


387 


Black-faced  numbers  refer  to  authors 


De  Morgan  and  the  actuary, 

945. 

Gauss  as  astronomer,  971. 
Laplace's  "It  is  easy  to  see." 

986. 
Lagrange,  Laplace  and  Gauss 

contrasted,  993. 
Newton's  interest  in  chemistry 

and  theology,  1015. 
On  Newton's  method  of  work, 

1026. 
On  Newton's  discovery  of  the 

calculus,  1027. 
Gauss's   estimate   of   Newton, 

1029. 

M.  and  philosophy,  1417. 
Advance  in  physics,  1530. 
Plato  on  geometry,  1804. 
Notation  of  the  calculus,  1904. 
Barnett,  M.  the  type  of  perfect 

reasoning,  307. 
Barrow,  On  the  method  of  m., 

213,  227. 

Eulogy  of  m.,  330. 
M.  as  a  discipline  of  the  mind, 

402. 

M.  and  eloquence,  830. 
Philosophy  and  m.,  1430. 
Uses  of  m.,  1572. 
On  surd  numbers,  1728. 
Euclid's  definition  of  propor- 
tion, 1835. 
Beattie,  1431. 

Beauty  of  m.,  453,  824, 1208. 
Consists    in    simplicity,    242, 

315. 

Sylvester  on,  1101. 
Russell  on,  1104. 
Young  on,  1110. 
Kummer  on,  1111. 
White  on,  1119. 
And  truth,  1114. 
Boltzmann  on,  1116. 
Beltrami,  On  reading  of  the  mas- 
ters, 614. 
Berkeley,  On  geometry  as  logic, 

428. 

On  math,  symbols,  1214. 
On  fluxions,   1915,  1942-1944. 
On   infinite   divisibility,    1945. 
Bernoulli,  Daniel,  919. 


Bernoulli,  James, 

Legend  for  his  tomb,  920,  922. 

Computation  of  sum  of  tenth 

powers    of    numbers,     921. 

Discussion  of  logarithmic  spiral, 

922. 
Berthelot,  M.  inspires  respect  for 

truth,  438. 
Bija  Ganita,  Solution  of  problems, 

1739. 
Billingsley,     M.     beautifies     the 

mind,  319. 

Binary  arithmetic,  991. 
Biology  and  m.,  1579-1581. 
Biot,  Laplace's  "  It  is  easy  to  see," 

986. 
Bocher,  M.  likened  to  painting, 

1103. 
Interrelation  of  m.  and  logic, 

1313. 
Geometry  as  a  natural  science, 

1866. 

Boerne,    On    Pythagoras,    1855. 
Bois-Reymond,  On  the  analytic 

method,  1893. 

Natural  selection  and  the  cal- 
culus, 1921. 
Boltzmann,    On    beauty    in  m., 

1116. 
Bolyai,  Janos, 

Duel  with  officers,  924. 
Universal  language,  925. 
Science  absolute  of  space,  926. 
Bolyai,  Wolfgang,  927. 

On  Gauss,  972. 
Bolzano,  928. 

Cured  by  Euclid,  929. 
Parallel  axiom,  2110. 
Book-keeping,  Importance  of  the 

art  of,  1571. 
Boole,  M.  E.  719. 
Boole's  Laws  of  Thought,  1318. 
Borda-Demoulins,        Philosophy 

and  m.,  1405. 
Boswell,  981. 
Bowditch,  On  Laplace's  "Thus  it 

plainly  appears,"  985. 
Boyle,  Usefulness  of  m.  to  physics, 

437. 

M.    and   science,    1513,    1533. 
Ignorance  of  m.,  1577. 


388 


INDEX 
Black-faced  numbers  refer  to  authors 


M.  and  physiology,  1582. 
Wings  of  m.,  1626. 
Advantages  of  algebra,    1703. 
Brahmagupta,    Estimate   of   m., 

320. 
Brewster,  On  Euler's  knowledge 

of  the  Aeneid,  959. 
On  Euler  as  a  computer,  963. 
On  Newton's  fame,  1002. 
Brougham,  1202. 
Buckle,  On  geometry,  1810, 1837. 
Burke,  On  the  value  of  m.,  447. 
Burkhardt,  On  discovery  in  m., 

618. 

On  universal  symbolism,  1221. 
Butler,  N.  M.,  M.  demonstrates 
the  supremacy  of  the  human 
reason,  309. 

M.  the  most  astounding  intel- 
lectual creation,  707. 
Geometry  before  algebra,  1871. 
Butler,  Samuel,  2118. 
Byerly,  On  hyperbolic  functions, 
1929. 

Cajori,  On  the  value  of  the  history 
of  m.,  615. 

On  Bolyai,  927. 

Cayley's  view  of  Euclid,  936. 

On  the  extent  of  Euler's  work, 
960. 

On  Euler's  math,  power,  964. 

On    the    Darmstaetter    prize, 
967. 

On   Sylvester's   first   class    at 
Johns  Hopkins,  1031. 

On  music  and  m.  among  the 
Pythagoreans,  1130. 

On  the  greatest  achievement  of 
the  Hindoos,  1615. 

On  modern  calculation,   1614. 

On  review  in  arithmetic,  1713. 

On  Indian  m.,  1737. 

On  the  characteristics  of  an- 
cient geometry,  1873. 

On  Napier's  rule,  1888. 
Calculating  machines,  1641. 
Calculation,  Importance  of,  602. 

Not  the  sole  object  of  m.,  268. 
Calculus,  Chapter  XIX.    Foun- 
dation of  253. 


As  a  method,  309. 

May  be  taught  at  an  early  age, 

519,  1917,  1918. 
Carlisle  life  tables,  946. 
Cambridge  m.,  836,  1210. 
Cantor,  On  freedom  in  m.,  205. 

207. 
On  the   character  of   Gauss's 

writing,  975. 
Zeno's  problem,  1938. 
On  the  infinite,  1952. 
Carnot,  On  limiting  ratios,  1908. 
On   the   infinitesimal   method, 

1907. 
Carson,    Value    of    geometrical 

training,  1841. 

Cartesian    method,    1889,    1890. 

Carus,    Estimate    of    m.,    326. 

M.  reveals  supernatural  God, 

460. 

Number  and  nature,  1603. 
Zero  and  infinity,  1948. 
Non-euclidean  geometry,  2016. 
Cathedral,      "  Petrified     mathe- 
matics," 1110. 
Causation  in  m.,  251,  254. 
Cayley,    Advantage    of    modern 
geometry  over  ancient,  711. 
On  the  imaginary,  722. 
Sylvester  on,  930. 
Noether  on,  931. 
His  style,  932. 
Forsyth  on,  932-934. 
His  method,  933. 
Compared  with  Euler,  934. 
Hermite  on,  935. 
His  view  of  Euclid,  936. 
His   estimate    of   quaternions, 

937. 

M.  and  philosophy,  1420. 
Certainty  of  m.,  222,  1440-1442, 

1628,  1863. 
Chamisso,  Pythagorean  theorem, 

1856. 

Chancellor,  M.  develops  observa- 
tion, imagination  and  reason, 
433. 
Chapman,    Different   aspects    of 

m.,  265. 

Characteristics  of  m.,  225,  229, 
247,  263. 


INDEX 
Black-faced  numbers  refer  to  authors 


389 


Characteristics    of    modern    m., 

720  724—729. 

Charm  in  m.,  1115,  1640,  1848. 
Chasles,    Advantage    of   modern 
geometry  over  ancient,  712. 
Checks  in  m.,  230. 
Chemistry  and  m.,   1520,    1560, 

1561,  1750. 
Chess,  M.  like,  840. 
Chrystal,  Definition  of  m.,  113. 
Definition  of  quantity,  115. 
On  problem  solving,  531. 
On    modern    text-books,    533. 
How  to  read  m.,  607. 
His  algebra,  635. 
On  Bernoulli's  numbers,   921. 
On   math,    versus   logical   ab- 

stractness,  1304. 
Rules  of  algebra,  1710. 
On  universal  arithmetic,  1717. 
On  Homer's  method,  1744. 
On  probabilities,  1967. 
Cicero,   Decadence   of   geometry 

among  Romans,  1807. 
Circle,  Properties  of,  1852,  1857. 
Circle-squarers,  2108,  2109. 
Clarke,     Descriptive     geometry, 

1882. 

Classic  problems,  Hilbert  on,  627. 
Clebsch,  On  math,  research,  644. 
Clifford,  On  direct  usefulness  of 

math,  results,  652. 
Correspondence     the     central 

idea  of  modern  m.,  726. 
His  vision,  938. 
His  method,  939. 
His   knowledge    of   languages, 

940. 

His  physical  strength,  941. 
On  Helmholtz,  979. 
On  m.  and  mineralogy,  1558. 
On  algebra  and  good  English, 

1712. 

Euclid  the  encouragement  and 
guide  of  scientific  thought, 
1820. 

Euclid  the  inspiration  and  as- 
piration of  scientific  thought, 
1821. 

On   geometry  for  girls,    1842. 
On  Euclid's  axioms,  2015. 


On     non-Euclidean   geometry, 

2022. 

Colburn,  967. 

Coleridge,    On   problems   in   m., 
534. 

Proposition,  gentle  maid,  1419. 

M.  the  quintessence  of  truth, 

2019. 
Colton,  On  the  effect  of  math. 

training,  417. 

Commensurable  numbers,  1966. 
Commerce  and  m.,  1571. 
Committee  of  Ten,  On  figures  in 
geometry,  524. 

On  projective  geometry,  1876. 
Common  sense,  M.  the  ethereali- 

zation  of,  312. 
Computation,  Not  m.,  515. 

And  m.,  810. 

Not    concerned    with    signifi- 
cance of  numbers,  1641. 
Comte,  On  the  object  of  m.,  103. 

On  the  business  of  concrete  m., 
104. 

M.  the  indispensable  basis  of  all 
education,  334. 

Mill  on,  942. 

Hamilton  on,  943. 

M.  and  logic,  1308,  1314,  1325. 

On  Kant's  view  of  m.,  1437. 

Estimate  of  m.,  1504. 

M.  essential  to  scientific  educa- 
tion, 1505. 

M.    and    natural    philosophy, 
1506. 

M.  and  physics,  1535, 1551. 

M.  and  science,  1536. 

M.   and   biology,    1578,    1580, 
1581. 

M.  and  social  science,  1687. 

Every  inquiry  reducible  to  a 
question  of  number,  1602. 

Definition  of  algebra  and  arith- 
metic, 1714. 

Geometry   a   natural    science, 
1813. 

Ancient  and  modern  methods, 
1875. 

On  the  graphic  method,  1881. 

On  descriptive  geometry,  1883. 

Mill's  estimate  of,  1903. 


390 


INDEX 
Black-faced  numbers  refer  to  authors 


Congreve,  2143. 
Congruence,  Symbol  of,  1646. 
Conic  sections,  658, 660,1541, 1542. 
Conjecture,   M.  free  from,  234. 
Contingent  truths,  1966. 
Controversies    in  m.,   215,   243, 

1859. 
Correlation  in  m.,  525-527,  1707, 

1710. 
Correspondence,  Concept  of,  725, 

726. 

Coulomb,  1516. 
Counting,  Every  problem  can  be 

solved  by,  1601. 

Cournot,  On  the  object  of  m.,  268. 
On  algebraic  notation,  1213. 
Advantage  of  math,  notation, 

1220. 
Craig,  On  the  origin  of  a  new 

science,  646. 
Credulity,  M.  frees  mind  from, 

450. 
Cremona,  On  English  text-books, 

609. 

Crofton, 

On  value  of  probabilities,  1590. 
On   probabilities,    1952,    1970, 

1972. 

Cromwell,  On  m.  and  public  serv- 
ice, 328. 

Curiosities,  Chapter  XXI. 
Curtius,  M.  and  philosophy,  1409. 
Curve,  Definition  of,  1927. 
Cyclometers,    Notions   of,   2108. 
Cyclotomy  depends  on  number 

theory,  1647. 

D'Alembert,  On  rigor  in  m.,  536. 

Geometry  as  logic,  1311. 

Algebra  is  generous,  1702. 

Geometrical    versus     physical 
truths,  1809. 

Standards  in  m.,  1851. 
Dante,  1858,  2117. 
Darmstaetter  prize,  2129. 
Davis,    On   Sylvester's   method, 
1035. 

M.  and  science,  1610. 

On  probability,  1968. 
Decimal    fractions,    1217,    1614. 
Decker,  2142. 


Dedekind,  Zeno's  Problem,  1938. 
Deduction,  Why  necessary,  219. 
M.  based  on,  224. 
And  Intuition,  1413. 
Dee,  On  the  nature  of  m.,  261. 
Definitions    of    m.,    Chapter    I. 

Also  2005. 
Democritus,  321. 
Demoivre,  His  death,  944. 
Demonstrations,  Locke  on,  236. 
Outside  of  m.,  1312. 
In  m.,  1423. 
De  Morgan,  Imagination  in  m., 

258. 
M.  as  an  exercise  in  reasoning, 

430. 

On  difficulties  in  m.,  521. 
On  correlation  in  m.,  525. 
On  extempore  lectures,  540. 
On    reading    algebraic    works, 

601. 

On  numerical  calculations,  602. 
On  practice  problems,  603. 
On  the  value  of  the  history  of 

m.,  615,  616. 
On  math'ns.,  812. 
On  Bacon's  knowledge  of  m., 

918. 

And  the  actuary,  945. 
On  life  tables,  946. 
Anagrams  on  his  name,  947. 
On  translations  of  Euclid,  953. 
Euclid's     elements    compared 
with     Newton's     Principia, 
954. 

Euler  and  Diderot,  966. 
Lagrange  and  the  parallel  axiom, 

984. 
Anagram  on  Macaulay's  name, 

996. 
Anagrams  on  Newton's  name, 

1028. 

On  math,  notation,  1216. 
Antagonism  of  m.   and  logic, 

1315. 

On  German  metaphysics,  1416. 
On  m.  and  science,  1537. 
On  m.  and  physics,  1538. 
On  the  advantages  of  algebra, 

1701. 
On  algebra  as  an  art,  1711. 


INDEX 


391 


Black-faced  numbers  refer  to  authors 


On  double  algebra  and  quater- 
nions, 1720. 
On  assumptions  in  geometry, 

1812. 

On    Euclid    in    schools,    1819. 
Euclid  not  faultless,  1823. 
On  Euclid's  rigor,  1831. 
Geometry  before  algebra,  1872. 
On  trigonometry,  1885. 
On  the  calculus  in  elementary 

instruction,  1916. 
On  integration,  1919. 
On  divergent  series,  1935,  1936. 
Ad  infinitum,  1949. 
On  the  fourth  dimension,  2032. 
Pseudomath  and  graphomath, 

2101. 

On  proof,  2102. 
On  paradoxers,  2105. 
Budget  of  paradoxes,  2106. 
On    D'Israeli's    six    follies    of 

science,  2107. 
On     notions     of     cyclometers 

2108. 

On  St.  Vincent,  2109. 
Where  Euclid  failed,  2114. 
On  the  number  of  the  beast, 

2151. 
Descartes,  On  the  use  of  the  term 

m.,  102. 
On    intuition    and    deduction, 

219,  1413. 
Math'ns  alone  arrive  at  proofs, 

817. 
The    most    completely    math. 

type  of  mind,  948. 
Hankel  on,  949. 
Mill  on,  950. 
Hankel  on,  1404. 
On  m.  and  philosophy,   1425, 

1434. 

Estimate  of  m.,  1426. 
Unpopularity  of,  1501. 
On  the  certainty  of  m.,  1628. 
On  the  method  of  the  ancients, 

1874. 

On  probable  truth,  1964. 
Descriptive    geometry,     1882, 

1883. 

Dessoir,  M.  and  medicine,  1585. 
Determinants,  1740,  1741. 


Diderot  and  Euler,  966. 
Differential      calculus,    Chapter 

XIX. 

And  scientific  physics,  1549. 
Differential  equations,  1549-1552, 

1924,  1926. 
Difficulties     in    m.,     240,    521, 

605-607,  634,  734,  735. 
Dillmann,  M.  a  royal  science,  204. 
On  m.  as  a  high  school  subject, 

401. 
Ancient  and  modern  geometry 

compared,  715. 
On  ignorance  of,  807. 
On  m.  as  a  language,  1204. 
Number   regulates   all   things, 

1505. 
Dirichlet,    On   math,    discovery, 

625. 

As  a  student  of  Gauss,  977. 
Discovery  in  m.,  617-622,  625. 
D-ism  versus  dot-age,  923. 
Disquisitiones  Arithmeticae,  975, 

977,  1637,  1638. 
D'Israeli,  2007. 
Divergent  series,  1935-1937. 
"  Divide  etimpera,"  631. 
Divine  character  of  m.,  325,  329. 
"Divinez  avant  de  demontrer," 

630. 

Division  of  labor  in  m.,  631,  632. 
Dodgson,     On    the    charm    of, 

302. 

Pythagorean    theorem,     1854. 
Ignes  fatui  in  m.,  2103. 
Dolbear,  On  experiment  in  math. 

research,  613. 
Domus    Lescinia,    Anagram    on, 

2155. 

Donne,  1816, 
Dot-age  versus  d-ism,  923. 
Durfee,  On  Sylvester's  forgetful- 
ness,  1038. 

Dutton,  On  the  ethical  value  of 
m.,  446. 

"Eadem  mutata  resurgo."    920, 

922. 
Echols,  On  the  ethical  value  of 

m.,  455. 
Economics  and  m.,   1593,   1594. 


392 


INDEX 


Black-faced  numbers  refer  to  authors 


Edinburgh  Review,  M.  and  as- 
tronomy, 1565,  1566. 
Education,  Place  of  m.  in,  334, 

408. 
Study  of  arithmetic  better  than 

rhetoric,  408. 
M.  as  an  instrument  in,  413, 

414. 

M.  in  primary,  431. 
M.  as  a  common  school  sub- 
ject, 432. 

Bain  on  m.  in,  442. 
Calculus  in  elementary,  1916, 

1917. 
Electricity,  M.  and  the  theory  of, 

1554. 

Elegance  in  m.,  640,  728. 
Ellis,  On  precocity  in  m.,  835. 
On   aptitude   of   Anglo-Danes 

for  m.,  836. 

On    Newton's    genius,     1014. 
Emerson,  On  Newton   and   La- 
place, 1003. 

On  poetry  and  m.,  1124. 
Endowment  of  math  ns,  818. 
Enthusiasm,  801. 
Equality,  Grassmann's  definition 

of,  105. 

Equations,  104,  526,  1891,  1892. 
Errors,  Theory  of,  1973,  1974. 
Esthetic  element  in  m.,  453^455, 

640, 1102, 1105,  1852, 1853. 
Esthetic  tact,  622. 
Esthetic  value  of  m.,  1848,  1850. 
Esthetics,  Relation  of  m.  to,  318, 

319,  439. 

Estimates    of   m.,    Chapter   III. 
See  also  1317,  1324,  1325,  1427, 

1504,  1508. 
Ethical  value  of  m.,  402,  438,  446, 

449,  455-457. 

Euclid,  Bolzano  cured  by,  929. 
And  Ptolemy,  951,  1878. 
And  the  student,  952. 
Euclidean    geometry,    711,    713, 

715. 

Euclid's  Elements, 
Translations  of,  953. 
Compared  with  the  Principia, 

954. 
Greatness  of,  955. 


Greatest    of    human    produc- 
tions, 1817. 

Performance  in,  1818. 

In  English  schools,  1819. 

Encouragement     and      guide, 
1820. 

Inspiration     and      aspiration, 
1821. 

The  only  perfect  model,  1822. 

Not  altogether  faultless,  1823. 

Only  a  small  part  of  m.,  1824. 

Not  fitted  for  boys,  1825. 

Early  study  of,  1826. 

Newton  and,  1827. 

Its  place,  1828. 

Unexceptional  in  rigor,    1829. 

Origin  of,  1831. 

Doctrine  of  proportion,   1834. 

Definition  of  proportion,  1835. 

Steps  in  demonstration,  1839. 

Parallel  axiom,  2007. 
Eudoxus,  904. 
Euler,  the  myriad-minded,  255. 

Pencil  outruns  intelligence,  626. 

On   theoretical   investigations, 
657. 

Merit  of  his  work,  956. 

The  creator  of  modern  math, 
thought,  957. 

His    general    knowledge,    958. 

His  knowledge  of  the  Aeneid, 
959. 

Extent  of  his  work,  960. 

"Analysis  incarnate,"  961. 

As  a  computer,  962,  963. 

His  math,  power,  964. 

His    Tentamen    novae    theorae 
musicae,  965. 

And  Diderot,  966. 

Error  in  Fermat's  law  of  prime 

numbers,  967. 
Eureka,  911,  917. 
Euripedes,  1568. 
Everett,    Estimate    of   m.,    325. 

Value  of  math,  training,  443. 

Theoretical  investigations,  656. 

Arithmetic  a  master-key,  1571. 

On  m.  and  law,  1598. 
Exactness,  See  precision. 
Examinations,  407. 
Examples,  422. 


INDEX 
Black-faced  numbers  refer  to  authors 


393 


Experiment  in  m.,  612,  613,  1530, 

1531 
Extent  of  m.,  737,  738. 

Fairbairn,  528. 
FaUacies,  610. 

Faraday,  M.  and  physics,  1554. 
Fermat,  255,  967,  1902. 
Fermat's  theorem,  2129. 
Figures,  Committee  of  Ten  on, 
524. 

Democritus  view  of,  321. 

Battalions  of,  1631. 
Fine  Art,  M.  as  a,  Chapter  XI. 
Fine,  Definition  of  number,  1610. 

On  the  imaginary,  1732. 
Fisher,  M.  and  economics,  1594. 
Fiske,   Imagination  in  m.,   256. 

Advantage  of  m.  as  logic,  1324. 
Fitch,  Definition  of  m.,  125. 

M.  in  education,  429. 

Purpose  of  teaching  arith- 
metic, 1624,  1625. 
Fizi,  Origin  of  the  Liliwati,  995. 
Flamsteed,  Anagram  on,  968. 
Fluxions,  1911,  1915,  1942-1944. 
Fontenelle,  Bernoulli's  tomb,  920. 
Formulas,  Compared  to  focus  of  a 

lens,  1515. 

Forsyth,  On  direct  usefulness  of 
math,  results,  654. 

On   theoretical   investigations, 
664. 

Progress  of  m.  704. 

On  Cayley,  932-934. 

On  m.  and  physics,  1539. 

On  m.  and  applications,  1540. 

On  invariants,  1747. 

On  function  theory,  1754,  1755. 
Foster,  On  m.  and  physics,  1516, 
1522. 

On    experiment    in   m.,    1531. 
Foundations  of  m.,  717. 
Four,  The  number,  2147,  2148. 
Fourier,  Math,  analysis  coexten- 
sive with  nature,  218. 

On  math,  research,  612. 

Hamilton  on,  969. 

On  m.  and  physics,  1552,  1553. 

On  the  advantage  of  the  Carte- 
sian method,  1889. 


Fourier's  theorem,  1928. 
Fourth  dimension,  2032,  2039. 
Frankland,  A.,  M.  and  chemistry, 

1560. 

Frankland,    W.    B.,    Motto    of 
Pythagorean      brotherhood, 
1833. 
The  most   beautiful  truth  in 

geometry,  1857. 
Franklin,    B.,    Estimate    of    m., 

322. 
On  the  value  of  the  study  of 

m.,  323. 

On  the  excellence  of  m.,  324. 
On   m.   as   a   logical   exercise, 

1303. 

Franklin,  F.,  On  Sylvester's  weak- 
ness, 1033. 

Frederick  the  Great,  On  geome- 
try, 1860. 

Freedom  in  m.,  205-208,  805. 
French  m.,  1210. 
Fresnel,  662. 
Frischlinus,  1801. 
Froebel,  M.  a  mediator  between 

man  and  nature,  262. 
Function  theory,  709,  1732,  1754, 

1755. 

Functional  exponent,  1210. 
Functionality,  The  central  idea  of 

modern  m.,  254. 
Correlated  to  life,  272. 
Functions,  1932,  1933. 

Concept  not  used  by  Sylvester, 

1034. 
Fundamental  concepts,   Chapter 

XX. 
Fuss,  On  Euler's  Tentamen  novae 

theorae  musicae,  965. 
Galileo,  On  authority  in  science, 

1528. 

Galton,  838. 
Gauss,  His  motto,  649. 
Mere  math'ns,  820. 
And  Newton  compared,  827. 
His  power,  964. 
His  favorite  pursuits,  970. 
The    first    of    theoretical    as- 
tronomers, 971. 

The  greatest  of  arithmeticians, 
971. 


394 


INDEX 
Black-faced  numbers  refer  to  authors 


The  math,  giant,  972. 
Greatness  of,  973. 
Lectures  to  three  students,  974. 
His  style  and  method,  983. 
His  estimate  of  Newton,  1029. 
On  the  advantage  of  new  cal- 
culi, 1216. 

M.  and  experiment,  1531. 
His  Disquisitiones  Arithmetical, 

1639,  1640. 
M.  the  queen  of  the  sciences, 

1642. 

On  number  theory,  1644. 
On  imaginaries,  1730. 
On  the  notation  sin^,  1886. 
On  infinite  magnitude,  1960. 
On    non-euclidean    geometry, 

2023-2028. 

On  the  nature  of  space,  2034. 
Generalization  in  m.,  245,  246, 

252,  253,  327,  728. 
Genius,  819. 
Geometrical  investigations,   642, 

643. 
Geometrical  training,   Value  of, 

1841,  1842,  1844-1846. 
Geometry,  Chapter  XVIII. 
Bacon's  definition  of,  106. 
Sylvester's  definition  of,  1 10. 
Value  to  mankind,  332,  449. 
And  patriotism,  332. 
An  excellent  logic,  428. 
Plato's  view  of,  429. 
The  fountain  of  all  thought, 

451. 

And  algebra,  525-527. 
Lack  of  concreteness,  710. 
Advantage    of    modern    over 

ancient,  711,  712. 
And  music,  965. 
And  arithmetic,  1604. 
Is  figured  algebra,  1706. 
Name  inapt,  1801. 
And  experience,  1814. 
Halsted's  definition  of,  1815. 
And  observation,  1830. 
Controversy  in,  1859. 
A  mechanical  science,  1865. 
A  natural  science,  1866. 
Not  an  experimental  science, 

1867. 


Should    come    before    algebra, 

1767,  1871,  1872. 
And  analysis,  1931. 
Germain,  Algebra  is  written  geom- 
etry, 1706. 
Gilman,  Enlist  a  great  math'n, 

808. 
Glaisher,  On  the  importance  of 

broad  training,  623. 
On  the  importance  of  a  well- 
chosen  notation,  634. 
On  the  expansion  of  the  field  of 

m.,  634. 
On  the  need  of  text-books  on 

higher  m.,  636. 
On    the    perfection    of    math. 

productions,  649. 
On  the  invention  of  logarithms, 

1616. 

On  the  theory  of  numbers,  1640. 
Goethe,  On  the  exactness  of  m., 

228. 
M.    an    organ    of   the   higher 

sense,  273. 
Estimate  of  m.,  311. 
M.  opens  the  fountain  of  all 

thought,  451. 
Math'ns  must  perceive  beauty 

of  truth,  803. 
Math'ns    bear    semblance    of 

divinity,  804. 

Math'ns  like  Frenchmen,  813. 
His  aptitude  for  m.,  976. 
M.  like  dialectics,  1307. 
On  the  infinite,  1957. 
Golden  age  of  m.,  701,  702. 

Of  art  and  m.  coincident,  1134. 
Gordan,  When  a  math,  subject  is 

complete,  636. 

Gow,  Origin  of  Euclid,  1832. 
Gower,  1808. 

Grammar  and  m.  compared,  441. 
Grandeur  of  m.,  325. 
Grassmann,  Definition  of  m.,  106. 
Definition  of  magnitude,   105. 
Definition  of  equality,  105. 
On  rigor  in  m.,  638. 
On  the  value  of  m.,  1512. 
Greek  view  of  science,  1429. 
Graphic  method,  1881. 
Graphomath,  2101. 


INDEX 


395 


Black-faced  numbers  refer  to  authors 


Group,  Notion  of,  1751. 
Growth  of  m.,  209,  211,  703. 

Hall,   G.   S.,   M.  the  ideal  and 
norm  of  all  careful  thinking, 
304. 
Hall  and  Stevens,  On  the  parallel 

axiom,  2008. 

Haller,  On  the  infinite,  1968. 
Halley,  On  Cartesian  geometry, 

716. 

Halsted,  On  Bolyai,  924-926. 
On  Sylvester,  1030,  1039. 
And  Slyvester,  1031,  1032. 
On  m.  as  logic,  1305. 
Definition  of  geometry,   1816. 
Hamilton,  Sir  William,  His  igno- 
rance of  m.,  978. 
Hamilton,  W.  R.,  Importance  of 

his  quaternions,  333. 
Estimate   of   Comte's   ability, 

943. 

To  the  memory  of  Fourier,  969. 
Discovery  in  light,  1558. 
On  algebra  as  the  science  of 

time,  1715,  1716. 
On  quaternions,  1718. 
On  trisection  of  an  angle,  2112. 
Hankel,  Definition  of  m.,  114. 
On  freedom  in  m.,  206. 
On  the  permanency  of  math. 

knowledge,  216. 
On  aim  in  m.,  608. 
On    isolated    theorems,     621. 
On  tact  in  m.,  622. 
On  geometry,  714. 
Ancient  and  modern  m.  com- 
pared, 718,  720. 
Variability  the  central  idea  in 

modern  m.,  720. 
Characteristics  of  modern  m., 

728. 

On  Descartes,  949. 
On  Euler's  work,  956. 
On  philosophy  and  m.,   1404. 
On  the  origin  of  m.,  1412. 
On  irrationals  and  imaginaries, 

1729. 

On  the  origin  of  algebra,  1736. 
Euclid  the  only  perfect  model, 
1822. 


Modern  geometry  a  royal  road, 

1878. 

Harmony,  326,  1208. 
Harris,  M.  gives  command  over 

nature,  434. 

Hathaway,  On  Sylvester,  1036. 
Heat,  M.  and  the  theory  of,  1552. 

1553. 
Heath,  Character  of  Archimedes' 

work,  913. 
Heaviside,  The  place  of  Euclid, 

1828. 

Hebrew  and  Latin  races,  Apti- 
tude for  m.,  838. 
Hegel,  1417. 
Heiss,   Famous  anagrams,   2055. 

Reversible  verses,  2056. 
Helmholtz,  M.  the  purest  form  of 

logical  activity,  231. 
M.  requires  perseverance  and 

great  caution,  240. 
M.  should  take  more  important 

place  in  education,  441. 
Clifford  on,  979. 
M.  the  purest  logic,  1302. 
M.  and  applications,  1445. 
On  geometry,  1836. 
On  the  importance  of  the  cal- 
culus, 1939. 

A  non-euclidean  world,   2029. 

Herbart,  Definition  of  m.,   117. 

M.    the   predominant   science, 

209. 
On   the   method    of    m.,   212, 

1576. 
M.  the  priestess  of  definiteness 

and  clearness,  217. 
On  the  importance  of  checks, 

230. 

On  imagination  in  m,  267. 
M.  and  invention,  406. 
M.  the  chief  subject  for  com- 
mon schools,  432. 
On  aptitude  for  m.,  609. 
On  the  teaching  of  m.,  516. 
M.  the  greatest  blessing,  1401. 
M.  and  philosophy,  1408. 
If  philosophers  understood  m., 

1415. 

M.    indispensable    to   science, 
1502. 


396 


INDEX 
Black-faced  numbers  refer  to  authors 


M.  and  psychology,  1683, 1684. 
On  trigonometry,  1884. 
Hermite,  On  Cayley,  935. 
Herschel,    M.    and    astronomy, 

1664. 

On  probabilities,  1692. 
Hiero,  903,  904. 
Higher  m.,  Mellor's  definition  of, 

108. 

Hilbert,  On  the  nature  of  m.,  266. 
On  rigor  in  m.,  537. 
On  the  importance  of  problems, 

624,  628. 
On  the  solvability  of  problems, 

627. 
Problems   should   be   difficult, 

629. 
On  the  abstract  character  of 

m.,  638. 

On  arithmetical  symbols,  1627. 
On    non-euclidean    geometry, 

2019. 

Hill,  Aaron,  On  Newton,   1009. 
Hill,  Thomas,   On  the  spirit  of 

mathesis,  274. 
M.  expresses  thoughts  of  God, 

275. 

Value  of  m.,  332. 
Estimate   of    Newton's   work, 

333. 

Math'ns  difficult  to  judge,  841. 
Math'ns  indifferent  to  ordinary 

interests  of  life,  842. 
A  geometer  must  be  tried  by 

his  peers,  843. 
On  Bernoulli's  spiral,  922. 
On  mathesis  and  poetry,  1125. 
On  poesy  and  m.,  1126. 
On   m.    as   a   language,    1209. 
Math,  language  untranslatable, 

1210. 

On  quaternions,  1719. 
On  the  imaginary,  1734. 
On    geometry    and    literature, 

1847. 

M.  and  miracles,  2167,  2158. 
Hindoos,   Grandest  achievement 

of,  1615. 

History  and  m.,  1599. 
History  of  m.,  615,  616,  625,  635. 
Hobson,  Definition  of  m.,  118. 


On  the  nature  of  m.,  252. 
Functionality  the  central  idea 

of  m.,  254. 
On   theoretical   investigations, 

663. 

On  the  growth  of  m.,  703. 
A  great  math'n  a  great  artist, 

1109. 

On  m.  and  science,  1508. 
Hoffman,  Science  and  poetry  not 

antagonistic,  1122. 
Holzmuller,  On  the  teaching  of 

m.,  518. 
Hooker,  1432. 
Hopkinson,  M.  a  mill,  239. 
Horner's  method,  1744. 
Howison,  Definition  of  m.,  134, 

135. 

Definition  of  arithmetic,  1612. 
Hudson,  On  the  teaching  of  m., 

512. 
Hughes,  On  science  for  its  own 

sake,  1546. 
Humboldt,    M.   and   astronomy, 

1667. 

Humor  in  m.,  539. 
Hume,    On    the    advantage    of 

math,  science,  1438. 
On  geometry,  1862. 
On  certainty  in  m.,  1863. 
Objection  to  abstract  reason- 
ing, 1941. 

Button,  On  Bernoulli,  919. 
On  Euler's  knowledge,  958. 
On    the    method    of    fluxions, 

1911. 
Huxley,  Negative  qualities  of  m., 

250. 

Hyperbolic  functions,  1929,  1930. 
Hyper-space,   2030,   2031,   2033, 
2036-2038. 

Ignes  fatui  in  m.,  2103. 
Ignorabimus,   None  in  m.,  627. 
Ignorance  of  m.,  310,  331,  807, 

1537,  1577. 

Imaginaries,  722,  1729-1735. 
Imagination  in  m.,  246,  251,  253, 

256-258,  433,  1883. 
Improvement  of  elementary  m., 

617. 


INDEX 
Black-faced  numbers  refer  to  authors 


397 


Incommensurable  numbers,  con- 
tingent truths  like,  1966. 
Indian  m.,  1736,  1737. 
Induction  in  m.,  220-223,  244. 

And  analogy,  724. 
Infinite  collection,  Definition  of, 

1959,  1960. 

Infinite  divisibility,  1945. 
Infinitesimal  analysis,  1914. 
Infinitesimals,    1905-1907,    1940, 

1946,  1954. 
Infinitum,  Ad,  1949. 
Infinity  and  infinite  magnitude, 

723,  928,   1947,  1948,  1950- 

1958. 
Integers,    Kronecker    on,    1634, 

1635. 
Integral  numbers,  Minkowsky  on, 

1636. 

Integrals,  Invention  of,  1922. 
Integration,     1919-1921,      1923, 

1925. 
International  Commission  on  m., 

501,  502,  938. 

Intuition  and  deduction,  1413. 
Isolated  theorems  in  m.,  620,  621. 
"It   is  easy  to  see,"  985,   986, 

1045. 
Invariance,    Correlated    to    life, 

272. 

MacMahon  on,  1746. 
Keyser  on,  1749. 
Invariants,    Changeless    in    the 

midst  of  change,  276. 
Importance  of  concept  of,  727. 
Sylvester  on,  1742. 
Forsyth  on,  1747. 
Keyser  on,  1748. 
Lie  on,  1752. 

Invention  in  m.,  251,  260. 
Inverse  process,  1207. 
Investigations,  See  research. 
Irrationals,  1729. 

Jacobi,  His  talent  for  philology, 
980. 

Aphorism,  1635. 

Die"EwigeZahl,"  1643. 
Jefferson,  Om  m.  and  law,  1597. 
Johnson,  His  recourse  to  m.,  981. 

Aptitude   for   numbers,    1617. 


On  round  numbers,  2137. 
Journals  and  transactions,  635. 
Jowett,  M.  as  an  instrument  in 

education,  413. 
Judgment,  M.  requires,  823. 
Jupiter's  eclipses,  1544. 
Justitia,  The  goddess,  824. 
Juvenal,     Nemo     mathematicus 

etc.,  831. 

Kant,  On  the  a  priori  nature  of 
m.,  130. 

M.    follows   the   safe   way   of 
science,  201. 

On  the  origin  of  scientific  m., 
201. 

On  m.  in  primary  education, 
431. 

M.  the  embarrassment  of  meta- 
physics, 1402. 

His  view  of  m.,    1436,   1437. 

On  the  difference  between  m. 
and  philosophy,  1436. 

On  m.  and  science,  1508. 

Esthetic  elements  in  m.,  1852, 
1853. 

Doctrine  of  time,  2001. 

Doctrine  of  space,  2003. 
Karpinsky,    M.    and    efficiency, 

1573. 
Kasner, 

"Divinez    avant    de    demon- 
trer,"  630. 

On  modern  geometry,  710. 
Kelland,   On   Euclid's   elements, 

1817. 

Kelvin,  Lord,  See  William  Thom- 
son. 
Kepler,  His  method,  982. 

Planetary  orbits  and  the  regu- 
lar solids,  2134. 
Keyser,  Definition  of  m.,  132. 

Three    characteristics    of    m., 
225. 

On  the  method  of  m.,  244. 

On  ratiocination,  246. 

M.  not  detached  from  life,  273. 

On  the  spirit  of  mathesis,  276. 

Computation  not  m.,  515. 

Math,  output  of  present  day, 
702. 


398 


INDEX 


Black-faced  numbers  refer  to  authors 


Modern   theory   of   functions, 

709. 

M.  and  journalism,  731. 
Difficulty  of  m.,  735. 
M.  appeals  to  whole  mind,  815. 
Endowment  of  math'ns,  818. 
Math'ns     in     public     service, 

823. 

The  aim  of  the  math'n,  844. 
On  Bolzano,  929. 
On  Lie,  992. 

On  symbolic  logic,  1321. 
On  the  emancipation  of  logic, 

1322. 
On  the  Principia  Mathematica, 

1326. 

On  invariants,  1728. 
On  invariance,  1729. 
On  the  notion  of  group,  1751. 
On    the    elements    of    Euclid, 

1824. 

On  projective  geometry,  1880. 
Definition    of    infinite    assem- 
blage, 1960. 
On  the  infinite,  1961. 
On    non-euclidean    geometry, 

2035. 

On  hyper-space,  2037,  2038. 
Khulasat-al-Hisab,  Problems, 

1738. 

Kipling,  1633. 
Kirchhoff,  Artistic  nature  of  his 

works,  1116. 

Klein,  Definition  of  m.,  123. 
M.  a  versatile  science,  264. 
Aim  in  teaching,  507,  517. 
Analysts     versus     synthesists, 

651. 

On  theory  and  practice,  661. 
Math,    aptitudes    of    various 

races,  838. 
Lie's  final  aim,  993. 
Lie's  genius,  994. 
On  m.  and  science,  1520. 
Famous  aphorisms,  1635. 
Calculating  machines,  1641. 
Calculus  for  high  schools,  1918. 
On  differential  equations,  1926. 
Definition  of  a  curve,  1927. 
On  axioms  of  geometry,  2006. 
On  the  parallel  axiom,  2009. 


On    non-euclidean    geometry, 

2017,  2021. 

On  hyper-space,  2030. 
Kronecker,  On  the  greatness  of 

Gauss,  973. 

God  made  integers  etc.,  1634. 
Kummer,  On  Dirichlet,  977. 
On  beauty  in  m.,  1111. 

LaFaille,    Mathesis    few    know, 

1870. 

Lagrange,  On  correlation  of  alge- 
bra and  geometry,  527. 

His  style  and  method,  983. 

And  the  parallel  axiom,  984. 

On  Newton,  1011. 

Wings  of  m.,  1604. 

Union  of  algebra  and  geometry, 
1707. 

On   the   infinitesimal   method, 

1906. 
Lalande,    M.   in    French  army, 

314. 

Langley,  M.  in  Prussia,  513. 
Lampe,  On  division  of  labor  in 
m.,  632. 

On  Weierstrass,  1049. 

Weierstrass      and     Sylvester, 
1050. 

Qualities  common  to  math'ns 
and  artists,  1113. 

Charm  of  m.,  1115. 

Golden  age  of  art  and  m.  coin- 
cident, 1134. 

Language,  Chapter  XII.  See 
also  311,  419,  443,  1523, 
1804,  1889. 

Laplace,  On  instruction  in  m., 
220. 

His  style  and  method,  983. 

"Thus    it    plainly    appears," 
985,  986. 

Emerson  on,  1003. 

On  Leibnitz,  991. 

On  the  language  of  analysis, 
1222. 

On  m.  and  nature,  1525. 

On  the  origin  of  the  calculus, 
1902. 

On  the  exactitude  of  the  differ- 
ential calculus,  1910. 


INDEX 


399 


Black-faced  numbers  refer  to  authors 


The  universe  in  a  single  for- 
mula, 1920. 
On    probability,     1963,    1969, 

1971. 
Laputa,  Math'ns  of,  2120-2122, 

Math,  school  of,  2123. 
Lasswitz,    On    modern    algebra, 

1741. 

On  function  theory,  1934. 
On  non-euclidean  geometry, 

2040. 

Latin  squares,  252. 
Latta,  On  Leibnitz's  logical  cal- 
culus, 1317. 

Law  and  m.,  1597,  1598. 
Laws  of  thought,  719,  1318. 
Leadership,   M.   as  training  for, 

317. 
Lefevre,    M.    hateful    to    weak 

minds,  733. 
Logic  and  m.,  1309. 
Leibnitz,  On  difficulties  in  m.,  241. 
His  greatness,  987. 
His  influence,  988. 
The  nature  of  his  work,  989. 
His  math,  tendencies,  990. 
His  binary  arithmetic,  991. 
On  Newton,  1010. 
On  demonstrations  outside  of 

m.,  1312. 

Ars  characteristica,  1316. 
His  logical  calculus,  1317. 
Union  of  philosophical  and  m. 

productivity,  1404. 
M.  and  philosophy,  1435. 
On    the    certainty    of    math. 

knowledge,  1442. 
On   controversy   in   geometry, 

1869. 

His  differential  calculus,  1902. 
His  notation  of  the  calculus, 

1904. 

On  necessary   and   contingent 
truth,  1966. 

Lecture,  Preparation  of,  540. 
Leverrier,  Discovery  of  Neptune, 

1559. 

Lewes,    On    the    infinite,    1953. 
Lie,    On   central   conceptions   in 

modern  m.,  727. 
Endowment  of  math'ns,   818. 


The    comparative    anatomist, 

992. 

Aim  of  his  work,  993. 
His  genius,  994. 
On  groups,  1752. 
On  the  origin  of  the  calculus, 

1901. 

On  differential  equations,  1924. 
Liliwati,  Origin  of,  995. 
Limitations    of    math,     science, 

1437. 
Limits,  Method  of,   1905,   1908, 

1909,  1940. 
Lindeman,   On  m.   and   science, 

1523. 

Lobatchewsky,  2022. 
Locke,    On   the   method   of   m., 

214,  235. 
On  proofs  and  demonstrations, 

236. 
On    the    unpopularity  of   m., 

271. 
On  m.   as   a  logical  exercise, 

423,  424. 

M.  cures  presumption,  425. 
Math,   reasoning  of  universal 

application,  426. 
On  reading  of  classic  authors, 

604. 

On  Aristotle,  914. 
On  m.  and  philosophy,   1433. 
On    m.    and    moral    science, 

1439,  1440. 
On    the    certainty    of    math. 

knowledge,  1440,  1441. 
On  unity,  1607. 
On  number,  1608. 
On  demonstrations  in  numbers, 

1630. 
On  the  advantages  of  algebra, 

1705. 

On  infinity,  1955,  1957. 
On  probability,  1965. 
Logarithmic  spiral,  922. 
Logarithmic  tables,  602. 
Logarithms,  1526,  1614,  1616. 
Logic  and  m.,  Chapter  XIII. 

See  also  423-430,  442. 
Logical  calculus,  1316,  1317. 
Longevity  of  math'ns,  839. 
Liouville,  822. 


400 


INDEX 
Black-faced  numbers  refer  to  authors 


Lovelace,  Why  are  wise  few  etc., 

1629. 
Lover,  2140. 

Macaulay,  Plato  and  Bacon,  316. 
On  Archimedes,  905. 
Bacon's  view  of  m.,  915,  916. 
Anagram  on  his  name,  996. 
Plato  and  Archytas,  1427. 
On  the  power  of  m.,  1527. 
Macfarlane,   On  Tait,   Maxwell, 

Thomson,  1042. 
On  Tait  and  Hamilton's  quater- 
nions, 1044. 
Mach,    On   thought-economy   in 

m.,  203. 

M.  seems  possessed  of  intelli- 
gence, 626. 

On  aim  of  research,  647. 
On  m.  and  counting,  1601. 
On   the   space   of   experience, 

2011. 

MacMahon,  Latin  squares,  252. 
On  Sylvester's  bend  of  mind, 

645. 

On  Sylvester's  style,  1040. 
On  the  idea  of  invariance,  1746. 
Magnitude,   Grassmann's  defini- 
tion, 105. 
Magnus,  On  the  aim  in  teaching 

m.,  505. 

Manhattan  Island,  Cost  of,  2130. 
Marcellus,    Estimate    of    Arch- 
imedes, 909. 
Maschke,    Man   above    method, 

650. 
Masters,  On  the  reading  of  the, 

614. 
Mathematic,  Sylvester  on  use  of 

term,  101. 

Bacon's  use  of  term,  106. 
Mathematical       faculty,       Fre- 
quency of,  832. 
Mathematical    mill,    The,    239, 

1891. 
Mathematical  productions,   648, 

649. 

Mathematical  theory,  When  com- 
plete, 636,  637. 

Mathematical       training,      443, 
444. 


Maxims  of  math'ns,  630,  631, 

649. 

Not  a  computer,  1211. 
Intellectual  habits  of  math'ns, 

1428. 

The  place  of  the,  1529. 
Characteristics  of  the  mind  of 

a,  1534. 
Mathematician,     The,     Chapter 

VIII. 
Mathematics,       Definitions     of, 

Chapter  I. 

Objects  of,  Chapter  I. 
Nature  of,  Chapter  II. 
Estimates  of,  Chapter  III. 
Value  of,  Chapter  IV. 
Teaching  of,  Chapter  V. 
Study  of,  Chapter  VI. 
Research  in,  Chapter  VI. 
Modern,  Chapter  VII. 
As  a  fine  art,  Chapter  XI. 
As  a  language,  Chapter  XII. 

Also  445,  1814. 
And  logic,  Chapter  XIII. 
And  philosophy,  Chapter  XIV. 
And  science,  Chapter  XV. 
And  applications,  Chapter  XV. 
Knowledge  most  in,  214. 
Suppl.  brevity  of  life,  218. 
The  range  of,  269. 
Compared  to  French  language, 

311. 

The  care  of  great  men,  322. 
And  professional  education,  429. 
And  science  teaching,  522. 
The  queen  of  the  sciences,  975. 
Advantage    over    philosophy, 

1436,  1438. 

As  an  instrument,  1506. 
For  its  own  sake,  1540,  1541, 

1545,  1546. 
The  wings  of,  1604. 
Mathesis,  274,  276,   1870,  2015. 
Mathews,  On  Disqu.  Arith.  1638. 
On  number  theory,  1639. 
The  symbol  = ,  1646. 
On  Cyclotomy,  1647. 
Laws  of  algebra,  1709. 
On  infinite,  zero,  infinitesimal, 

1954. 
Maxwell,  1043,  1116. 


INDEX 
Black-faced  numbers  refer  to  authors 


401 


Maxims  of  great  math'ns,  630, 

631,  649. 

McCormack,  On  the  unpopular- 
ity of  m.,  270. 
On  function,  1933. 
Me'chanique  celeste,  985,  986. 
Medicine,  M.  and  the  study  of, 

1585,  1918. 
Mellor,  Definition  of  higher  m., 

108. 

Conclusions  involved  in  prem- 
ises, 238. 

On  m.  and  science,  1561. 
On  the  calculus,  1912. 
On  integration,  1923,  1926. 
Memory  in  m.,  253. 
Menaechmus,  901. 
Mere  math'ns,  820,  821. 
Merz,  On  the  transforming  power 

of  m.,  303. 
On  the  dominant  ideas  in  m., 

725. 

On  extreme  views  in  m.,  827. 
On  Leibnitz's  work,  989. 
On    the    math,    tendency    of 

Leibnitz,  990. 
On  m.  as  a  lens,  1515. 
M.   extends  knowledge,   1524. 
Disquisitiones       Arithmeticae, 

1637. 

On  functions,  1932. 
On  hyper-space,  2036. 
Metaphysics,  M.  the  only  true, 

305. 

Meteorology  and  m.,  1557. 
Method  of  m.  212-215,  226,  227, 

230,  235,  244,  806,  1576. 
Metric  system,  1725. 
Military  training,  M.  in,  314,  418, 

1574. 

Mill,  On  induction  in  m.,  221, 222. 
On  generalization  in  m.,  245. 
On  math,  studies,  409. 
On  m.  in  a  scientific  education, 

444. 

Math'ns  hard  to  convince,  811. 
Math'ns  require  genius,   819. 
On  Comte,  942. 
On  Descartes,  942,  948. 
On    Sir    William    Hamilton's 
ignorance  of  m.,  978. 


On  Leibnitz,  987. 
On  m.  and  philosophy,  1421. 
On  m.  as  training  for  philos- 
ophers, 1422. 
M.    indispensable  to    science, 

1519. 

M.  and  social  science,  1595. 
On   the   nature   of   geometry, 

1838. 

On  geometrical  method,  1861. 
On  the  calculus,  1903. 
Miller,     On    the    Darmstaetter 

prize,  2129. 
Milner,    Geometry    and    poetry, 

1118. 
Minchin,  On  English  text-books, 

539. 

Mineralogy  and  m.,  1558. 
Minkowski,  On  integral  numbers, 

1636. 

Miracles  and  m.,  2157, 2158, 2160. 
Mixed  m.,  Bacon's  definition  of, 

106. 

Whewell's   definition   of,    107. 
Modern  m.,  Chapter  VII. 
Modern  algebra,  1031, 1032, 1638, 

1741. 
Modern    geometry,     1710-1713, 

715,  716,  1878. 
Moebius,   Math'ns  constitute  a 

favorite  class,  809. 
M.  a  fine  art,  1107. 
Moral  science  and  m.,  1438-1440. 
Moral  value  of  m.,  See  ethical 

value. 
Mottoes,  Of  math'ns,  630,  631, 

649. 

Of  Pythagoreans,  1833. 
Murray,  Definition  of  m.,  116. 
Music   and   m.,    101,   276,    965, 
1107,  1112,  1116,  1127,  1128, 
1130-1133,  1135,  1136. 
Myers,  On  m.  as  a  school  subject, 

403. 

On  pleasure  in  m.,  454. 
On   the  ethical  value  of  m., 

457. 
On   the   value   of   arithmetic, 

1622. 

Mysticism  and   numbers,  2136- 
2141,  2143. 


402 


INDEX 
Black-faced  numbers  refer  to  authors 


Napier's  rule,  1888. 
Napoleon,  M.  and  the  welfare  of 
the  state,  313. 

His  interest  in  m.,  314,  1001. 
Natural  science  and  m.,  Chapter 
XV. 

Also  244,  444,  445,  501. 
Natural  selection,  1921. 
Nature  of  m.,  Chapter  II. 

See  also  815,  1215,  1308,  1426, 

1525,  1628. 
Nature,  Study  of,  433^36,  514, 

516,  612. 

Navigation  and  m.,  1543,  1544. 
Nelson,  Anagram  on,  2153. 
Neptune,    Discovery    of,     1554, 

1559. 
Newcomb,  On  geometrical  para- 

doxers,  2113. 
Newton, 

Importance  of  his  work,  333. 

On  correlation  in  m.,  626. 

On  problems  in  algebra,  530. 

And  Gauss  compared,  827. 

His  fame,  1002. 

Emerson  on,  1003. 

Whewell  on,  1004,  1005. 

Arago  on,  1006. 

Pope  on,  1007. 

Southey  on,  1008. 

Hill  on,  1009. 

Leibnitz  on,  1010. 

Lagrange  on,  1011. 

No  monument  to,  1012. 

Wilson  on,  1012,  1013. 

His  genius,  1014. 

His  interest  in  chemistry  and 
theology,  1015. 

And  alchemy,  1016,  1017. 

His  first  experiment,  1018. 

As  a  lecturer,  1019. 

As  an  accountant,  1020. 

His  memorandum-book,  1021. 

His  absent-mindedness,  1022. 

Estimate  of  himself,  1023-1025. 

His  method  of  work,  1026. 

Discovery  of  the  calculus,  1027. 

Anagrams  on,  1028. 

Gauss's  estimate  of,  1029. 

On  geometry,  1811. 

Compared  with  Euclid,  1827. 


Geometry  a  mechanical  science, 
1865. 

Test  of  simplicity,  1892. 

Method  of  fluxions,  1902. 
Newton's  rule,  1743. 
Nile,  Origin  of  name,  2150. 
Noether,  On  Cayley,  931. 

On  Sylvester,  1034,  1041. 
Non-euclidean    geometry,    1322, 
2016-2029,  2033,  2035,  2040. 
Nonnus,    On    the    mystic    four, 

2148. 
Northrup,     On     Lord     Kelvin, 

1048. 

Notation,    Importance    of,    634, 
1222,  1646. 

Value  of  algebraic,  1213,  1214. 

Criterion  of  good,  1216. 

On  Arabic,  1217,  1614. 

Advantage  of  math.,  1220. 

See  also  symbolism. 
Notions,  Cardinal  of  m.,  110. 

Indefinable,  1219. 
Novalis,   Definition  of  pure  m., 
112. 

M.    the    life    supreme,    329. 

Without    enthusiasm    no    m., 
801. 

Method  is  the  essence  of  m, 
806. 

Math'ns  not  good  computers, 
810. 

Music  and  algebra,  1128. 

Philosophy  and  m.,  1406. 

M.  and  science,  1507, 1526. 

M.  and  historic  science,  1599. 

M.  and  magic,  2159. 

M.  and  miracles,  2160. 
Number,  Every  inquiry  reducible 
to  a  question  of,  1602. 

And  nature,  1603. 

Regulates  all  things,  1605. 

Aeschylus  on,  1606. 

Definition  of,  1609, 1610. 

And  superstition,  1632. 

Distinctness  of,  1707. 

Of  the  beast,  2151,  2152. 
Number-theory, 

The  queen  of  m.,  975. 

Nature  of,  1639. 

Gauss  on,  1644. 


INDEX 


403 


Black-faced  numbers  refer  to  authors 


Smith  on,  1645. 

Notation  in,  1646. 

Aid  to  geometry,  1647. 

Mystery  in,  1648. 
Numbers,   Pythagoras'   view  of, 
321. 

Mighty  are,  1568. 

Aptitude  for,  1617. 

Demonstrations  in,  1630. 

Prime,  1648. 

Necessary  truths  like,  1966. 

Round,  2137. 

Odd,  2138-2141. 

Golden,  2142. 

Magic,  2143. 
Number-work,  Purpose  of,  1623. 

Obscurity  in  m.  and  philosophy, 

1407. 
Observation  in  m.,  251-253,  255, 

433,  1830. 

Obviousness  in  m.,  985,  986,  1045. 
Olney,  On  the  nature  of  m.,  253. 
Oratory  and  m.,  829,  830. 
Order  and  arrangement,  725. 
Origin  of  m.,  1412. 
Orr,  Memory  verse  for  «",  2127. 
Osgood,  On  the  calculus,  1913. 
Ostwald,     On     four-dimensional 

space,  2039. 

IT.  In  actuarial  formula,  945. 

Memory  verse  for,  2127. 
Pacioli,   On   the   number   three, 

2145. 

Painting  and  m.,  1103,  1107. 
Papperitz,  On  the  object  of  pure 

m.,  111. 

Paradoxes,  Chapter  XXL 
Parallel    axiom,    Proof    of,    984, 

2110,  2111. 

See  also  non-euclidean  geom- 
etry. 
Parker,  Definition  of  arithmetic, 

1611. 
Number  born  in  superstition, 

1632. 

On  geometry,  1805. 
Parton,  On  Newton,  1917-1919, 

1021,  1022,  1827. 
Pascal,  Logic  and  m.,  1306. 


Peacock,    On   the   mysticism   of 

Greek  philosphers,  2136. 
The  Yankos  word  for  three, 

2144. 

The  number  of  the  beast,  2152. 
Pearson,  M.  and  natural  selection, 

834. 
Peirce,   Benjamin,   Definition   of 

m.,  120. 

M.  as  an  arbiter,  210. 
Logic  dependent  on  m.,  1301. 
On  the  symbol  x/— 1,  1733. 
Peirce,   C.  S.   Definition  of  m., 

133. 

On  accidental  relations,  2128. 
Perry,  On  the  teaching  of  m.,  510, 

511,  519,  837. 
Persons  and  anecdotes,  Chapters 

IX  and  X. 

Philosophy  and  m.,  Chapter  XIV. 
Also  332,  401,  414,  444,  445, 

452. 
Psychology  and  m.,  1576,  1583, 

1584. 

Physics  and  m.,  129,  437,  1516, 
1530,  1535,  1538,  1539,  1548, 
1549,  1550,  1555,  1556. 
Physiology  and  m.,  1578,  1581, 

1582. 
Picard,  On  the  use  of  equations, 

1891. 

Pierce,  On  infinitesimals,  1940. 
Pierpont,    Golden    age    of    m., 

701. 

On  the  progress  of  m.,  708. 
Characteristics  of  modern  m., 

717. 

On  variability,  721. 
On  divergent  series,  1937. 
Plato,  His  view  of  m.,  316,  429. 
M.  a  study  suitable  for  free- 
men, 317. 

His  conic  sections,  332. 
And  Archimedes,  904. 
Union    of   math,    and   philos- 
ophical productivity,  1404. 
Diagonal  of  square,  1411. 
And  Archytas,  1427. 
M.  and  the  arts,  1567. 
On  the  value  of  m.,  1574. 
On  arithmetic,  1620, 1621. 


404 


INDEX 
Black-faced  numbers  refer  to  authors 


God  geometrizes,   1635,   1636. 

1702. 
On  geometry,  429,  1803,  1804, 

1806,  1844,  1845. 
Pleasure,  Element  of  in  m.,  1622, 

1629,  1848,  1850,  1851. 
Pliny,  2039. 

Plus  and  minus  signs,  1727. 
Plutarch,   On   Archimedes,    903, 

904,  908-910,  912. 
God  geometrizes,  1802. 
Poe,  417. 
Poetry  and  m., 
Weierstrass  on,  802. 
Pringsheim  on,  1108. 
Wordsworth  on,  1117. 
Milner  on,  1118. 
Workman  on,  1120. 
Pollock  on,  1121. 
Hoffman  on,  1122. 
Thoreauon,  1123. 
Emerson  on,  1124. 
Hill  on,  1125, 1126. 
Shakespeare  on,  1127. 
Poincar6,  On  elegance  in  m.,  640. 
M.    has    a    triple    end,    1102. 
M.  as  a  language,  1208. 
Geometry  not  an  experimental 

science,  1867. 

On  geometrical  axioms,  2005. 
Point,  1816. 

Politics,  Math'ns  and,  814. 
Political  science,  M.  and,  1201, 

1324. 
Pollock,    On    Clifford,    938-941, 

1121. 

Pope,  907,  2015,  2031,  2046. 
Precision  in  m.,  228,  639,  728. 
Precocity  in  m.,  835. 
Predicabilia  a  priori,  2003. 
Press,  M.  ignored  by  daily,  731, 

732. 
Price,  Characteristics  of  m.,  247. 

On  m.  and  physics,  1650. 
Prime   numbers,    Sylvester    on, 

1648. 

Principia  Mathematica,  1326. 
Pringsheim,  M.  the  science  of  the 

self-evident,  232. 
M.  should  be  studied  for  its  own 
sake,  439. 


On  the  indirect  value  of  m., 

448. 

On  rigor  in  m.,  635. 
On  m.  and  journalism,  732. 
On  math'ns  in  public  service, 

824. 
Math'n  somewhat  of  a  poet, 

1108. 

On  music  and  m.,  1132. 
On  the  language  of  m.,  1211. 
On  m.  and  physics,  1648. 
Probabilities,     442,    823,     1589, 
1590-1592,  1962-1972,  1975. 
Problems,  In  m.,  523,  534. 
In  arithmetic,  528. 
In  algebra,  530. 
Should  be  simple,  603. 
In  Cambridge  texts,  608. 
On  solution  of,  611. 
On  importance  of,  624,  628. 
What  constitutes  good,  629. 
Aid  to  research,  644. 
Of  modern  m.,  1926. 
Problem  solving,  531,  532. 
Proclus,    Ptolemy    and    Euclid, 

951. 
On  characteristics  of  geometry, 

1869. 
Progress  in  m.,  209,  211,  212,  216, 

218,  702-705,  708. 
Projective  geometry,  1876,  1877, 

1879,  1880. 
Proportion,  Euclid's  doctrine  of, 

1834. 

Euclid's    definition    of,    1835. 
Proposition,  1219,  1419. 
Prussia,  M.  in,  513. 
Pseudomath,  Defined,  2101. 
Ptolemy  and  Euclid,  951. 
Publications,    Math,    of   present 

day,  702,  703. 
Public  service,  M.  and,  823,  824, 

1303,  1574. 
Public   speaking,    M.   and,   420, 

829,  830. 
Pure  M.,  Bacon's  definition  of, 

106. 

Whewell's  definition  of,  107. 
On  the  object  of,  111,  129. 
Novalis'  conception  of,  112. 
Hobson's  definition  of,  118. 


INDEX 


405 


Black-faced  numbers  refer  to  authors 


Russell's  definition  of,  127,  128. 
Pursuit  of  m.,  842. 
Pythagoras, 

Number  the  nature  of  things, 

321. 

Union  of  math,  and  philosophi- 
cal productivity,  1404. 
The  number  four,  2147. 
Pythagorean  brotherhood,  Motto 

of,  1833. 
Pythagoreans,    Music    and    M., 

1130. 

Pythagorean  theorem,  1854-1856, 
2026. 

Quadrature,  See  Squaring  of  the 

circle. 
Quantity,  Chrystal's  definition  of, 

116. 

Quarles,  On  quadrature,  2116. 
Quaternions,  333,  841,  937,  1044, 

1210,  1718-1726. 
Quetelet,  Growth  of  m.,  1514. 

Railway-making,  1570. 
Reading  of  m.,  601,  604-606. 
Reason,  M.  most  solid  fabric  of 

human,  308. 
M.  demonstrates  supremacy  of 

human,  309. 
Reasoning,  M.  a  type  of  perfect, 

307. 
M.  as  an  exercise  in,  423-427, 

429,  430,  1503. 
Recorde,    Value    of    arithmetic, 

1619. 

Regiomontanus,  1543. 
Regular  solids,  2132-2135. 
Reid,   M.   frees  from  sophistry, 

215. 
Conjecture  has  no  place  in  m., 

234. 

M.  the  most  solid  fabric,  308. 
On  Euclid's  elements,  955. 
M.  manifests  what  is  impossible 

1414. 

On  m.  and  philosophy,  1423. 
Probability    and    Christianity, 

1975. 

On  Pythagoras  and  the  regular 
solids,  2132. 


Reidt,  M.  as  an  exercise  in  lan- 
guage, 419. 
On   the   ethical   value  of   m., 

456. 
On  aim  in  math,  instruction, 

506. 
Religion  and  m.,  274-276,  459, 

460,  1013. 

Research  in  m.,  Chapter  VI. 
Reversible  verses,  2156. 
Reye,  Advantages  of  modern  over 

ancient  geometry,  714. 
Rhetoric  and  m.,  1599. 
Riemann,    On   m.    and   physics, 

1549. 

Rigor  in  m.,  535-538. 
Rosanes,  On  the  unpopularity  of 

m.,  730. 

Royal  road,  201,  901,  951,  1774. 
Royal  science,  M.  a,  204. 
Rudio,  On  Euler,  957. 
M.  and  great  artists,  1105. 
On  m.  and  navigation,  1543. 
Rush,  M.  cures  predisposition  to 

anger,  458. 
Russell,   Definition   of  m.,    127, 

128. 
On    nineteenth    century    m., 

705. 
Chief  triumph  of  modern  m., 

706. 

On  the  infinite,  723. 
On  beauty  in  m.,  1104. 
On  the  value  of  symbols,  1219. 
On  Boole's  Laws  of  Thought, 

1318. 

Principia  Mathematica,  1326. 
On  geometry  and  philosophy, 

1410. 

Definition  of  number,  1609. 
Fruitful   uses    of    imaginaries, 

1735. 
Geometrical  reasoning  circular, 

1864. 

On  protective  geometry,  1879. 
Zeno's  problems,  1938. 
Definition  of  infinite  collection, 

1959. 

On  proofs  of  axioms,  2013. 
On    non-euclidean    geometry, 
2018. 


406 


INDEX 
Black-faced  numbers  refer  to  authors 


Safford,  On  aptitude  for  m.,  620. 

On  m.  and  science,  1509. 
Sage,  Battalions  of  figures,  1631. 
Sartorius,  Gauss  on  the  nature  of 

space,  2034. 
Scepticism,  452,  811. 
Schellbach,  Estimate  of  m.,  306. 

On    truth,    1114. 
Schiller,     Archimedes    and    the 

youth,  907. 

Schopenhauer,    Arithmetic    rests 
on  the  concept  of  time,  1613. 
Predicabilia  a  priori,  2003. 
Schroder,  M.  as  a  branch  of  logic, 

1323. 
Schubert,  Three  characteristics  of 

m.,  229. 

On  controversies  in  m.,   243. 
Characteristics  of  m.,  263. 
M.  an  exclusive  science,  734. 
Science  and  m.,  Chapter  XV. 
M.  an  indispensible  tool  of,  309. 
Neglect  of  m.  works  injury  to, 

310. 

Craig  on  origin  of  new,  646. 
Greek  view  of,  1429. 
Six  follies  of,  2107. 
See  also  433,  436,  437,  461,  725. 
Scientific  education,  Math,  train- 
ing   indispensable  basis   of, 
444. 

Screw,  The  song  of  the,  1894. 
As  an  instrument  in  geometry, 

2114. 
Sedgwick,  Quaternion  of  maladies, 

1723. 

Segre,   On  research  in  m.,   619. 
What    kind    of    investigations 

are  important,  641. 
On  the  worthlessness  of  certain 

investigations,  642,  643. 
On  hyper-space,  2031. 
Seneca,  Alexander  and  geometry, 

902. 
Seventy-seven,   The   number, 

2149. 

Shakespeare,    1127,    1129,    2141. 
Shaw,   J.   B.,   M.   like  game  of 

chess,  840. 

Shaw,  W.  H.,  M.  and  professional 
life,  1696. 


Sherman,  M.  and  rhetoric,  1699. 

Smith,  Adam,  1324. 

Smith,  D.  E.,  On  problem  solving, 

632. 
Value  of  geometrical  training, 

1846. 
Reason  for  studying  geometry, 

1860. 
Smith,  H.  J.  S.,  When  a  math. 

theory  is  completed,  637. 
On  the  growth  of  m.,    1521. 
On  m.  and  science,  1542. 
On  m.  and  physics,  1666. 
On  m.  and  meteorology,  1557. 
On  number  theory,  1646. 
Rigor  in  Euclid,  1829. 
On  Euclid's  doctrine  of  propor- 
tion, 1834. 
Smith,  W.  B.,  Definition  of  m., 

121. 

On  infinitesimal  analysis,  1914. 
On  non-euclidean  and  hyper- 
spaces,  2033. 
Simon,    On    beauty    and    truth, 

1114. 

Simplicity  in  m.,  315,  526. 
Sin20,  On  the  notation  of,  1886. 
Six  hundred  sixty-six,  The  num- 
ber, 2151,  2152. 
Social  science  and  m.,  1201,  1586, 

1587. 
Social  service,  M.  as  an  aid  to, 

313,  314,  328. 

Social  value  of  m.,  456,  1588. 
Solitude  and  m.,  1849,  1851. 
Sound,  M.  and  the  theory  of, 

1551. 

Sophistry,  M.  free  from,  215. 
Southey,  On  Newton,  1008. 
Space,  Of  experience,  2011. 
Kant's  doctrine  of,  2003. 
Schopenhauer's       predicabilia, 

2004. 

Whewell,  On  the  idea  of,  2004. 
Non-euclidean,      2015,     2016, 

2018. 
Hyper-,  2030,  2031,  2033, 2036- 

2038. 
Speer,  On  m.  and  nature-study, 

514. 
Spence,  On  Newton,  1016,  1020. 


INDEX 


407 


Black-faced  numbers  refer  to  authors 


Spencer,  On  m.  in  the  arts,  1570. 
Spedding,  On  Bacon's  knowledge 

of  m.,  917. 

Spherical  trigonometry,  1887. 
Spira  mirabilis,  922. 
Spottiswoode,  On  the  kingdom  of 

m.,  269. 
Squaring  the  circle,  1537,   1858, 

1934,  1948,  2115-2117. 
St.   Augustine,   The    number 

seventy  seven,  2149. 
Steiner,  On  projective  geometry, 

1877. 

Stewart,  M.  and  facts,  237. 
On  beauty  in  m.,  242. 
What  we  most  admire  in  m., 

315. 

M.  for  its  own  sake,  440. 
M.  the  noblest  instance  of  force 

of  the  human  mind,  452. 
Math'ns  and  applause,  816. 
Mere  math'ns,  821. 
Shortcomings       of      math  'ns, 

828. 
On  the  influence  of  Leibnitz, 

988. 

Reason  supreme,  1424. 
M.  and  philosophy  compared, 

1428. 
M.    and    natural    philosophy, 

1555. 
Stifel,  The  number  of  the  beast, 

2152. 
Stobaeus,    Alexander    and     Me- 

nsechmus,  901. 

Euclid  and  the  student,   952. 
Study  of  m.,  Chapter  VI. 
St.  Vincent,  As  a  circle-squarer, 

2109. 

Substitution,  Concept  of,  727. 
Superstition,  M.  frees  mind  from, 

450. 

Number  was  born  in,  1632. 
Surd  numbers,  1728. 
Surprises,  M.  rich  in,  202. 
Swift,  On  m.  and  politics,  814. 
The  math'ns  of  Laputa,  2120- 

2122. 
The  math,  school  of  Laputa, 

2123. 
His  ignorance  of  m.,  2124,  2125. 


Sylvester,  On  the  use  of  the  terms 

mathematic      and      mathe- 
matics, 101. 
Order    and    arrangement    the 

basic  ideas  of  m.,  109,  110. 
Definition  of  algebra,  110. 
Definition  of  arithmetic,  110. 
Definition  of  geometry  ,110. 
On  the  object  of  pure  m.,  129. 
M.  requires  harmonious  action 

of  all  the  faculties,  202. 
Answer  to  Huxley,  251. 
On  the  nature  of  m.,  251. 
On  observation  in  m.,  255. 
Invention  in  m.,  260. 
M.  entitled  to  human  regard, 

301. 

On  the  ethical  value  of  m.,  449. 
On  isolated  theorems,  620. 
"Auge  et  impera."    631. 
His  bent  of  mind,  645. 
Apology  for  imperfections,  648. 
On   theoretical   investigations, 

658. 
Characteristics  of  modern  m., 

724. 
Invested  m.  with  halo  of  glory, 

740. 

M.  and  eloquence,  829. 
On  longevity  of  math'ns,  839. 
On  Cayley,  930. 
His  view  of  Euclid,  936. 
Jacobi's   talent  for   philology, 

980. 

His  eloquence,  1030. 
Researches  in  quantics,   1032. 
His  weakness,  1033,  1036, 1037. 
One-sided     character     of     his 

work,  1034. 

His  method,  1035,  1036,  1041. 
His  forgetfulness,   1037,   1038. 
Relations  with  students,  1039. 
His  style,  1040,  1041. 
His  characteristics,  1041. 
His  enthusiasm,  1041. 
The  math.  Adam,  1042. 
And  Weierstrass,  1050. 
On  divine  beauty  and  order  in 

m.,  1101. 

M.  among  the  fine  arts,  1106. 
On  music  and  m.,  1131. 


408 


INDEX 
Black-faced  numbers  refer  to  authors 


M.    the   quintessence   of   lan- 
guage, 1205. 

M.  the  language  of  the  universe, 
1206. 

On  prime  numbers,  1648, 

On  determinants,  1740. 

On  invariants,  1742. 

Contribution  to  theory  of  equa- 
tions, 1743. 

To    a    missing    member    etc., 
1746. 

Invariants      and     isomerism, 
1760. 

His  dislike  for  Euclid,  1826. 

On  the  invention  of  integrals, 
1922. 

On  geometry  and  analysis,  1931. 

On  paradoxes,  2104. 
Symbolic    language,    M.    as    a, 
1207,  1212. 

Use  of,  1573. 

Symbolic  logic,  1316-1321. 
Symbolism,    On    the    nature    of 
math.,  1210. 

Difficulty  of  math.,  1218. 

Universal  impossible,  1221. 

See  also  notation. 
Symbols,  M.  leads  to  mastery  of, 
421. 

Value   of   math.,    1209,    1212, 
1219. 

Essential    to     demonstration, 
1316. 

Arithmetical,  1627. 
Symbols,  Burlesque  on,  1741. 

Tact  in  m.,  622,  623. 
Tait,  On  the  unpopularity  of  m., 
740. 

And  Thomson,  1043. 

And  Hamilton,  1044. 

On  quaternions,  1724-1726. 

On     spherical     trigonometry, 

1887. 

Talent,  Math'ns  men  of,  825. 
Teaching  of  m.,  Chapter  V. 
Tennyson,  1843. 
Text-books,  Chrystal  on,  533. 

Minchin  on,  539. 

Cremona  on  English,  609. 

Glaisher  on  need  of,  635. 


Thales,  201. 

Theoretical   investigations,   652- 

664. 

Theory  and  practice,  661. 
Teutonic  race,  Aptitude  for  m., 

838. 

Thompson,  Sylvanus,  Lord  Kel- 
vin's definition  of  a  math'n, 
822. 

Cayley's   estimate   of   quater- 
nions, 937. 
Thomson's  "  It  is  obvious  that," 

1046. 
Anecdote  of  Lord  Kelvin,  1046, 

1047. 
On  the  calculus  for  beginners, 

1917. 

Thomson,  Sir  William, 
M.  the  only  true  metaphysics, 

306. 
M.  not  repulsive  to  common 

sense,  312. 

What  is  a  math'n?  822. 
And  Tait,  1043. 
"It  is  obvious  that,"  1045. 
Anecdotes    concerning,     1046, 

1047,  1048. 

On  m.  and  astronomy,  1562. 
On  quaternions,  1721,  1722. 
Thomson  and  Tait,  1043. 

On  Fourier's  theorem,  1928. 
Thoreau,  On  poetry  and  m.,  1123. 
Thought-economy    in    m.,    203, 

1209,  1704. 

Three,  The  Yankos  word  for,  2 144. 

Pacioli  on  the  number,  2145. 

Time,  Arithmetic  rests  on  notion 

of  1613. 
As  a  concept  in  algebra,  1715, 

1716,  1717. 

Kant's  doctrine  of,  2001. 
Schopenhauer's  predicabilia, 

2003. 
Todhunter,  On  m.  as  a  university 

subject,  405. 
On  m.  as  a  test  of  performance, 

408. 

On  m.  as  an  instrument  in  edu- 
cation, 414. 

M.  requires  voluntary  exertion, 
415. 


INDEX 
Black-faced  numbers  refer  to  authors 


409 


On  exercises,  422. 

On  problems,  623,  608. 

How  to  read  m.,  605,  606. 

On  discovery  in  elementary  m., 
617. 

On  Sylvester's  theorem,  1743. 

On  performance  in  Euclid,  1818. 
Transformation,  Concept  of,  727. 
Trigonometry,   1881,    1884-1889. 
Trilinear  co-ordinates,  611. 
Trisection  of  angle,  2112. 
Truth,  and  m.,  306. 

Math'ns  must  perceive  beauty 
of,  803. 

And  beauty,  1114. 
Tzetzes,  Plato  on  geom.,  1803. 

Unity,  Locke  on  the  idea  of,  1607. 

Universal  algebra,  1753. 

Universal  arithmetic,  1717. 

Universal  language,  925. 

Unpopularity  of  m.,  270,  271, 
730-736, 738, 740, 1501, 1628. 

Usefulness,  As  a  principle  in  re- 
search, 652-655,659,  664. 

Uses  of  m.,  See  value  of  m. 

Value  of  m.,  Chapter  IV. 

See  also  330,  333,  1414,  1422, 
1505,  1506,  1512,  1523,  1526, 
1527,  1533,  1541,  1542,  1543, 
1547-1576,  1619-1626,  1841, 
1844-1851. 
Variability,  The  central  idea  of 

modern  m.,  720,  721. 
Venn,    On    m.    as    a    symbolic 

language,  1207. 
M.  the  only  gate,  1617. 
Viola,  On  the  use  of  fallacies,  610. 
Virgil,  2138. 

Voltaire,  Archimedes  more  imag- 
inative than  Homer,  259. 
M.  the  staff  of  the  blind,  461. 
On  direct  usefulness  of  results, 

663. 

On  infinite  magnitudes,  1947. 
On  the  symbol,  1950. 
Anagram  on,  2154. 

Walcott,  On  hyperbolic  functions, 
1930. 


Walker,   On  problems  in  arith- 
metic, 528. 
On  the  teaching  of  geometry, 

529. 
Wallace,  On  the  frequency  of  the 

math,  faculty,  832. 
On  m.  and  natural  selection, 

833,  834. 
Parallel    growth    of    m.    and 

music,  1135. 

Walton,  Angling  like  m.,  739. 
Weber,  On  m.  and  physics,  1549. 
Webster,   Estimate   of  m.,   331. 
Weierstrass,  Math'ns  are  poets, 

802. 

Anecdote  concerning,  1049. 
And  Sylvester,  1050. 
Problem  of  infinitesimals,  1938. 
Weismann,  On  the  origin  of  the 

math,  faculty,  1136. 
Wells,  On  m.  as  a  world  language, 

1201. 
Whately,  On  m.  as  an  exercise, 

427. 

On  m.  and  navigation,   1644. 
On     geometrical     demonstra- 
tions, 1839. 
On   Swift's   ignorance   of   m., 

2124. 
Whetham,    On    symbolic    logic, 

1319. 
Whewell,   On  mixed    and    pure 

math.,  107. 
M.  not  an    inductive  science, 

223. 

Nature  of  m.,  224. 
Value  of  geometry,  445. 
On   theoretical   investigations, 

660,  662. 

Math'ns  men  of  talent,   825. 
Fame  of  math'ns,  826, 
On  Newton's  greatness,  1004. 
On  Newton's  theory,  1005. 
On  Newton's  humility,  1025. 
On  symbols,  1212. 
On  philosophy  and  m.,  1429. 
On  m.  and  science,  1534. 
Quotation  fromR.  Bacon,  1547. 
On  m.  and  applications,  1541. 
Geometry      and      experience, 
1814.  " 


410 


INDKX 


Black-faced  numbers  refer  to  authors 


Geometry    not    an    inductive 

science,  1830. 
On  limits,  1909. 
On  the  idea  of  space,  2004. 
On    Plato    and    the    regular 

solids,  2133,  2135. 
White,  H.  S.,  On  the  growth  of  m., 

211. 
White,  W.  F.,  Definition  of  m., 

131,  1203. 
M.  as  a  prerequisite  for  public 

speaking,  420. 
On  beauty  in  m.,  1119. 
The  place  of  the  math'n,  1529. 
On  m.  and  social  science,  1586. 
The  cost  of  Manhattan  island, 

2130. 
Whitehead,  On  the  ideal  of  m., 

119. 

Definition  of  m.,  122. 
On  the  scope  of  m.,  126. 
On  the  nature  of  m.,  233. 
Precision  necessary  in  m.,  639. 
On  practical  applications,  655. 
On   theoretical   investigations, 

659. 

Characteristics  of  ancient  geom- 
etry, 713. 

On  the  extent  of  m.,  737. 
Archimedes  compared  with 

Newton,  911. 

On  the  Arabic  notation,  1217. 
Difficulty   of  math,   notation, 

1218. 

On  symbolic  logic,  1320. 
Principia  Mathematica,  1326. 
On  philosophy  and  m.,  1403. 
On  obscurity  in  m.  and  philos- 
ophy, 1407. 

On  the  laws  of  algebra,  1708. 
On  +  and  —  signs,  1727. 
On  universal  algebra,  1753. 
On  the  Cartesian  method,  1890. 
On   Swift's   ignorance   of   m., 

2125. 


Whitworth,   On  the  solution  of 

problems,  611. 
Williamson,  On  the  value  of  m., 

1575. 

Infinitesimals  and  limits,  1905. 
On  infinitesimals,  1946. 
Wilson,  E.  B.,  On  the  social  value 

of  m.,  1588. 

On  m.  and  economics,  1593. 
On    the    nature     of    axioms, 

2012. 

Wilson,   John,   On  Newton  and 
Shakespeare,  1012. 
Newton  and  Linnaeus,  1013. 
Woodward,      On      probabilities, 

1589. 
On  the  theory  of  errors,  1973, 

1974. 
Wordsworth,  W.,  On  Archimedes, 

906. 
On  poetry  and  geometric  truth, 

1117. 

On  geometric  rules,  1418. 
On  geometry,  1840,  1848. 
M.  and  solitude,  1859. 
Workman,  On  the  poetic  nature 
of  m.,  1120. 

Young,  C.  A.,  On  the  discovery  of 

Neptune,  1559. 
Young,  C.  W.,  Definition  of  m., 

124. 
Young,  J.  W.  A.,  On  m.  as  type  a 

of  thought,  404. 
M.  as  preparation  for  science 

study,  421. 
M.  essential  to  comprehension 

of  nature,  435. 

Development  of  abstract  meth- 
ods, 729. 

Beauty  in  m.,  1110. 
On  Euclid's  axiom,  2014. 

Zeno,  His  problems,  1938. 
Zero,  1948,  1954. 


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